text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
\section{Introduction}
Relational or interaction variables are common feature of modern data sets, and these are often represented as a network. Examples include friendships or communication within a social network, regulatory interactions among genes, transportation between cities, and relations or hyperlinks in information systems. Many, perhaps most of these systems are also dynamic in nature, and their evolving structure is commonly represented as a sequence of graphs~\cite{clauset2007persistence,berger2010dynamic,gauvin2014detecting,kim2013nonparametric,mucha2010community, rossi2011modeling,xing2010state,zhu2014scalable}.
Recently, a variety of techniques have been developed for automatically detecting communities---a task that is similar to traditional clustering~\cite{von2012clustering}, but on graphs---in these dynamic networks. These techniques include variants of multilayer or temporal modularity optimization~\cite{mucha2010community, bassett2013robust, bazzi2014community}, non-negative matrix or tensor factorization~\cite{acar2009link, dunlavy2011temporal, gauvin2014detecting, rossi2011modeling,zhu2014scalable}, minimum description length~\cite{sun2007graphscope,rosvall2010mapping}, and probabilistic models~\cite{yang2009bayesian,xing2010state,kim2013nonparametric,xu2014dynamic,han2014consistent,peixoto2015inferring,valles2014multilayer}. See Refs.~\cite{aggarwal2014evolutionary, hartmann2014clustering} for reviews. Despite these advances, relatively little is known about their optimality or the fundamental difficulty of detecting community structure in dynamic networks. In this paper, we derive a mathematically precise threshold on the detectability of communities in dynamic networks and give two algorithms that are optimal in the sense that they succeed all the way down to this threshold.
Community detection in dynamic networks inherits many of the challenges of community detection in static networks, including learning the number of communities, their sizes and node membership, and the pattern of connections among communities, e.g., assortative, disassortative, core-periphery, etc. It also poses new challenges, because both the network edges and the community memberships may evolve over time. A common approach is to simply take the union of dynamic graphs over a certain time window, and treat the resulting graph with techniques from static network analysis~\cite{clauset2007persistence}, thereby ignoring the dynamics within the window. Here, we explicitly model the dynamic nature of these networks and the way community memberships change over time, integrating information about the communities in an optimal way.
Our approach relies on probabilistic generative models, which can be used to learn latent community structure in real networks via Bayesian inference and to generate synthetic networks with known structure that can be used as benchmarks. A number of such models have recently been proposed for detecting communities in dynamic networks~\cite{kim2013nonparametric,xu2014dynamic}, including those based on the stochastic block model (SBM)~\cite{han2014consistent,yang2009bayesian} and its mixed membership counterpart~\cite{xing2010state}. Indeed, the variant of the stochastic block model~\cite{holland1983stochastic, nowicki2001estimation} we analyze here is a special case of some of these models:\ namely, where nodes change their community membership over time, but where edges are generated independently at each time step. As a result, the network of connections between nodes at different times is locally treelike, which makes a belief-propagation approach asymptotically optimal and allows us to compute the detectability threshold exactly.
In static networks, it has recently been shown that there exists a phase transition in the detectability of communities~\cite{decelle2011asymptotic,mossel2012reconstruction,massoulie2014community,mossel-neeman-sly-colt14} such that below the transition \textit{no algorithm can recover the true communities better than chance} (for two groups of equal size) but that efficient algorithms exist above it. Here, we generalize this result to dynamic networks, deriving a mathematically precise expression that describes where the detectability transition occurs as a function of both the strength of the communities and how quickly their membership is changing. When temporal correlations in community membership are present, we show that community detection in dynamic networks improves substantially over detection in static networks (or in a dynamic network where we cluster each graph independently).
Finally, we give two principled and efficient algorithms for community detection in dynamic networks. Specifically, we use belief propagation (BP) to pass messages between neighbors both within a given graph and between time-adjacent graphs to integrate information over the network's history in an optimal way. We then linearize BP to obtain a spectral algorithm, based on a dynamic version of the non-backtracking matrix~\cite{Krzakala2013,bordenave2015non}. We show experimentally that these algorithms can accurately recover the true community structure in dynamic networks all the way down to the threshold.
\section{A dynamic stochastic block model}
\label{sec:generative}
The stochastic block model (SBM) is a classic model of community structure in static networks. Here, we use a variant of the SBM in which the community labels of nodes change over time, but where edges are independent, which is a special case of several models previously introduced for community detection in dynamic networks~\cite{kim2013nonparametric,xu2014dynamic,han2014consistent,yang2009bayesian,xing2010state}. Crucially, our variant captures the important behavior of changing community labels and is analytically tractable.
Under the SBM a graph $G=(V, E)$ is generated as follows. Using a prior distribution $q_r$ over $k$ group or community labels, we assign each of the $n$ nodes $i \in V$ to a group $g_i$. We then generate the edges $E$ according to the probability specified by a $k \times k$ community interaction matrix $p$ and the group assignments $g$. In the sparse case, where $|E|=O(n)$, the resulting network is locally tree like and the number of edges between groups is Poisson distributed with parameter $c_{rs} = np_{rs}$.
In a dynamic network, we have a sequence of graphs $G(t)=(V, E(t))$ with $0 \le t \le T$, where each graph has its own group assignment vector $\left\{ g_i(t)\,|\, i\in V, t\in\{1,\ldots,T\}\right\}$. To generate each such assignment, we draw $g_i(0)$ from the prior, where each node has probability $q_r$ of being in community $1 \le r \le k$. With probability $\eta$, each node keeps its label from one time step to the next $g_i(t)=g_i(t-1)$, and otherwise it chooses a new label $g_i(t)$ from the prior $q_r$. Formally, the transition probability for community memberships is
\begin{equation}
P( g(t) \,|\, g(t-1) )= \prod_{i} \left( \eta\, \delta_{g_i(t),g_i(t-1)} + (1-\eta) q_{g_i(t)}\right) \enspace,
\label{eq:tran_prob}
\end{equation}
where $\delta_{a,b}=1$ if $a=b$ and 0 otherwise. The edges $E(t)$ are then generated independently for each $t$ according to the community interaction matrix $p$, by connecting each pair of nodes at the same time $i(t)$ and $j(t)$ with probability $p_{g_i(t),g_j(t)}$. Note that while the group assignments may change over time, the matrix $p$ remains constant. Subsequently, we use $A^{(t)}$ to denote the adjacency matrix for the graph $(V,E(t))$ at time $t$, and $D^{(t)}$ to denote the diagonal matrix of node degrees at time $t$, i.e. $D^{(t)}_{uv} = \delta_{uv} \sum_w A^{(t)}_{uw}$.
At successive times in this model, edges are correlated only through the group assignments $\{g(t)\}$. Given these, the full likelihood of a graph sequence under this dynamic SBM is
\begin{align}
P&( \{ E(t) \},\{ g(t) \}\,|\, p,\eta) =
P(\{ g(t)\})\prod_{t=0}^{T}\left(\prod_{(i,j) \in E(t)} p_{g_i(t),g_j(t)} \prod_{(i,j) \notin E(t)} 1-p_{g_i(t),g_j(t)}\right) \enspace ,
\label{eq:tsbm_lik}
\end{align}
where $P(\{ g(t)\})=P(g(0))\prod_{t=1}^TP(g(t)\,|\,g(t-1))$.
For our subsequent analysis, we focus on the common choices of a uniform prior $q_r = 1/k$, and where $c_{rs}=np_{rs}$ has two distinct entries:\ $c_{rs}=c_\mathrm{in}$ if $r=s$ and $c_{rs}=c_\mathrm{out}$ if $r\ne s$. In this setting the average degree of each graph is then $ c = \frac{1}{k}[{c_\mathrm{in} + (k-1) c_\mathrm{out}}]$. We are interested in the sparse regime where $c = O(1)$, because most real-world networks of interest are sparse (e.g., the Facebook social network), and sparsity allows us to carry out asymptotically optimal inference. Note that the case where every group has distinct average degrees is easier than the equal-average-degree case that we consider, because distinct average degrees give prior information about group memberships.
\section{The detectability threshold in dynamic networks}
\label{sec:detectability}
The fundamental question we now consider is, under what conditions can we detect, better than chance, the correct time-evolving labeling of the latent communities in this model?
Previous work on community detection in static networks has shown that there exists a sharp threshold below which no algorithm can perform better than chance in recovering the latent community structure~\cite{decelle2011asymptotic,mossel2012reconstruction}, at least in the case $k=2$. This threshold occurs at positive values of the difference in the internal and external group connection probabilities, meaning that the community structure may still exist, but is undetectable. In terms of the SBM's parameters, this phase transition occurs at
\begin{equation}\label{eq:static:thre}
|c_\mathrm{in} - c_\mathrm{out}| = k \sqrt{c} \enspace .
\end{equation}
In a dynamic network where community memberships correlate across time, we will exploit these correlations to improve upon the static detectability threshold. In the worst case, when these temporal correlations are absent, i.e., $\eta=0$, we should do no worse than the static threshold.
To facilitate our analysis, we define an extended graph structure, called a \textit{spatiotemporal graph}, in which we take $G(t)$ and add special ``temporal'' edges that connect each node $i(t)$ with its time-adjacent versions $i(t-1)$ and $i(t+1)$. Under our model, the ``spatial'' edges $E(t)$ are independent and sparse, implying that this spatiotemporal graph is locally treelike.
Consider a particular node $i(t)$ as $n\to\infty$ and $T\to\infty$. Moving outward in space and time, inference becomes a tree reconstruction problem, with stochastic transition matrix~$\sigma$, along each spatial edge
\begin{equation}
\sigma
= \lambda \mathbb{I} + (1-\lambda) \frac{J}{k} \enspace ,
\end{equation}
where $\mathbb{I}$ is the identity matrix, $J$ is the matrix of all $1$s, and
\begin{equation}
\lambda = \frac{c_\mathrm{in}-c_\mathrm{out}}{kc} \enspace .
\end{equation}
Similarly, along each temporal edge we have a stochastic matrix
\begin{equation}
\tau = \eta \mathbb{I} + (1-\eta) \frac{J}{k} \enspace .
\end{equation}
Thus, moving along a spatial or temporal edge copies a community label with probability $\lambda$ or $\eta$ respectively, and otherwise randomizes it according to the prior. That is, these edges multiply the distribution of labels by the stochastic matrices $\sigma$ and $\tau$, whose eigenvalues are $\lambda$ and $\eta$, other than the trivial eigenvalue $1$ corresponding to the uniform distribution.
Since each node in the spatiotemporal graph has $\mathrm{Poi}(c)$ (Poisson-distributed random variable with mean $c$) spatial edges but exactly two temporal edges, the tree is generated by a two-type branching process. Each spatial edge gives rise to two temporal edges (to each of the time-adjacent versions of its end point), and each temporal edge gives rise to one temporal edge (continuing in the same direction in time), and both give rise to $\mathrm{Poi}(c)$ spatial edges. Thus the matrix describing the expected number of children (where we multiply a column vector of populations on the left) is
$\begin{pmatrix}
c & c \\
2 & 1
\end{pmatrix} \, $.
Using the results of Ref.~\cite{janson2004robust}, the detectability threshold occurs when the largest eigenvalue of matrix $\begin{pmatrix} c \lambda^2 & c \lambda^2 \\ 2 \eta^2 & \eta^2 \end{pmatrix} \,$
exceeds unity, which yields
\begin{equation}
\label{eq:threshold}
c \lambda^2 > \frac{1-\eta^2}{1+\eta^2} \enspace .
\end{equation}
When $\eta = 0$, i.e., when there is no temporal correlation in community assignments over time, Eq.~\eqref{eq:threshold} recovers the static detectability threshold $c \lambda^2 > 1$, which is equivalent to Eq.~\eqref{eq:static:thre}. On the other hand, when $\eta = 1$, i.e., when the community assignments are fixed across time, we may simply integrate the graph over $T$, making it arbitrarily dense. We then have detectability for any $\lambda > 0$, implying that any amount of community structure can be detected. At intermediate values of $\eta$, the detectability threshold falls between these two extremes.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.80\columnwidth]{Fig_1.pdf}
\end{center}
\vspace{-8mm}
\caption{A schematic representation of belief propagation messages (see Eqs.\eqref{eq:spatial0} and~\eqref{eq:temporal0}) being passed along spatial and temporal edges in a spatiotemporal graph.\label{fig:msg_passing}
}
\end{figure}
This analysis corresponds to \emph{robust reconstruction} on trees, where we are given noisy information at the leaves of a tree and we want to propagate this information to the root~\cite{janson2004robust}. For $k=2$ groups, it is known rigorously in the static case~\cite{mossel2012reconstruction} that detecting the communities below this bound is information-theoretically impossible. We conjecture that the same is true in the dynamic case. For $k > 4$ groups, it has been conjectured~\cite{decelle2011asymptotic} that it is information-theoretically possible to succeed beyond the Kesten-Stigum bound, but that doing so takes exponential time.
\section{Bayesian inference of the model}
\label{sec:inference}
Given an observed graph sequence $G(t)$, we use Bayesian inference to learn the posterior distribution of latent community assignments:
\begin{equation}
P( \{ g(t) \}\,|\,\{ E(t) \}, p,\eta) =\frac{ P( \{ E(t) \},\{ g(t) \}\,|\, p,\eta)}{\sum_{\{\gamma(t)\}}P( \{ E(t) \},\{ \gamma(t) \}| p,\eta)}.
\end{equation}
This distribution is hard to compute in general because the summation runs over an exponential number of terms. However, when the spatiotemporal graph is sparse, as generated by our model, we may make a controlled Bethe approximation (also known as belief propagation (BP) in machine learning and as the ``cavity method'' in statistical physics) that allows us to carry out Bayesian inference in an efficient and asymptotically optimal way. We now describe a BP algorithm for learning our model form data, which we then linearize to obtain a fast spectral approach, based on a dynamic version of the non-backtracking matrix. This yields two inference algorithms that perform accurately all the way down to the transition.
\subsection{Belief propagation}
\label{BP_section}
Instead of inferring the joint posterior distribution, we use belief propagation to compute posterior marginal probabilities of node labels $\{\mu_s^i(t)\}$ over time. Belief propagation assumes conditional independence of these marginals, which is exact when the graph is a tree and is a good approximation when the graph is locally-tree like, as in our spatiotemporal graph. In our setting, nodes update their current belief about marginals according to the marginals of both their spatial and temporal neighbors. That is, we define two types of messages:\ \textit{spatial} messages that pass along spatial edges and \textit{temporal} messages that pass along temporal edges. Fig.~\ref{fig:msg_passing} illustrates this message passing scheme for a spatiotemporal graph.
A spatial message $\mu^{i \to j}_r(t)$ gives the marginal probability of a node $i$ at time $t$ being in community $r$, when we consider node $j$ to be absent at time $t$. This message is computed as
\begin{align}
\mu^{i \to j}_r(t)
&= \frac{q_r}{Z^{i \to j}(t)}
\left(\eta \mu^{i(t-1) \to i(t)}_r + (1-\eta) \sum_u \mu^{i(t-1) \to i(t)}_u q_u \right)\nonumber\times \prod_{\substack{\ell: (i,\ell) \in E(t) \\ \ell \ne j}} \sum_s c_{rs} \mu^{\ell \to i}_s(t)
\;\\
&\times\left(\eta \mu^{i(t+1) \to i(t)}_r + (1-\eta) \sum_u \mu^{i(t+1) \to i(t)}_u q_u \right) \times \;\prod_{\substack{\ell: (i,\ell) \notin E(t) \\ \ell \ne j}} \sum_s (1-p_{rs}) \mu^{\ell \to i}_s(t)
\enspace ,
\label{eq:spatial0}
\end{align}
where $Z^{i\to j}(t)$ is the normalization. The temporal message $\mu^{i(t) \to i(t+1)}_r$ (or $\mu^{i(t) \to i(t-1)}_r$) represents the marginal probability of node $i$ at time $t$ being in community $r$, when we consider node $i$ to be absent at time $t+1$ (or at $t-1$) and has a similar form:
\begin{align}
\mu^{i(t) \to i(t\pm 1)}_r
&= \frac{q_r}{Z^{i(t) \to i(t\pm 1)}}
\left(\eta \mu^{i(t\mp 1) \to i(t)}_r + (1-\eta) \sum_u \mu^{i(t\mp 1) \to i(t)}_u q_u \right) \nonumber \\
&\times \prod_{\ell: (i,\ell) \in E(t)} \sum_s c_{rs} \mu^{\ell \to i}_s(t)
\;\times \;\prod_{\ell: (i,\ell) \notin E(t)} \sum_s (1-p_{rs}) \mu^{\ell \to i}_s(t)
\enspace .
\label{eq:temporal0}
\end{align}
When $t=0$ or $t=T$, we remove the term corresponding to the temporal edge coming from outside the domain of $t$. Furthermore, following past work on BP for the static SBM~\cite{decelle2011asymptotic,aicher2014wsbm}, we exploit these networks' sparsity to reduce the computational complexity of the spatial updates at the cost of introducing $o(\frac{1}{n})$ corrections in sparse graphs. Specifically, we let $\mu^{\ell \to i}_s(t)$ be the same for all of $\ell$'s non-neighbors $i$. We then model the effects of all such non-edges as an adaptive external field on each node, which depends on the current estimated marginals $\mu^\ell_s(t)$. That is, we let
\mbox{$ \prod_{\ell} \sum_s (1-p_{rs}) \mu^\ell_s(t) \approx \mathrm{e}^{-h_r(t)}$} ,
where
\mbox{$ h_r(t) = \frac{1}{n} \sum_s c_{rs} \sum_\ell \mu^\ell_s(t)$} ,
which has the effect of preventing belief propagation from putting all the nodes at a given time into the same community. The adaptive fields only need to be updated after each BP iteration. This approximation yields a significant improvement in efficiency, reducing the computational complexity to be proportional to total number of edges in the spatiotemporal graph $c nT$, rather than $n^{2}T$.
Once the BP messages converge, we compute the marginal probability $\mu^i_r(t)$ that node $i$ belongs $r$ at time $t$. This is identical to~\eqref{eq:spatial0} and~\eqref{eq:temporal0}, except that we take all incoming edges into account. We then obtain a partition by marginalization, which assigns each node to its most-likely group:
\begin{equation}
\hat g_i(t)=\text{argmax}_{r}\mu^{i}_r(t) \enspace .
\end{equation}
It is well known in Bayesian inference~\cite{iba1999nishimori} that if the marginals are exact, then the marginalized partition is the optimal estimator of the latent community labels. Because spatiotemporal graphs under our model are sparse, we know that with $n\to\infty$, the marginals given by BP are asymptotically correct. Thus, our BP algorithm succeeds all the way down to the detectability threshold given by Eq.~\eqref{eq:threshold}, and gives an asymptotically optimal partition in terms of accuracy.
\subsection{Spectral clustering}
\label{sec:spectral}
The BP equations described above can be linearized to obtain a fast spectral approach for detecting community structure in dynamic networks. It is easy to verify that in our setting, when $q_r=1/k$, the average degree in each group is $c$. This implies that BP equations will always have a solution
\begin{equation}
\mu_r^{i(t)\to j(t)}=\mu_s^{i(t\pm 1)\to i(t)}=\frac{1}{k} \enspace ,
\end{equation}
which we call a \textit{factorized fixed point}. This fixed point only reflects the permutation symmetry in the system, and could be unstable due to random perturbations. If we use the correct parameters in BP equations, i.e., the same parameter used to generate the observed network, then in the language of physics we would say that system is in the Nishimori line~\cite{iba1999nishimori}. That is, if the BP messages deviate from the factorized solution, then they are correlated with the latent community labels and we say that there is no spin glass phase in system~\cite{iba1999nishimori}. This allows us to simplify the BP equations by studying how the messages deviate from the factorized solution, which results in a linearized version of BP. In the static SBM, this linearization is equivalent to a spectral clustering algorithm using the non-backtracking matrix~\cite{Krzakala2013}.
To do this, we rewrite the BP messages $\mu_r^{i(t)\to j(t)}$ as the uniform
fixed point $\frac{1}{k}$ plus deviations away from it. The vector of deviations is given by
\begin{align}
\Delta^{i(t)\to j(t)}=\left\{\mu_1^{i(t)\to j(t)},\mu_2^{i(t)\to j(t)},...,\mu_k^{i(t)\to j(t)}\right\}-\left\{\frac{1}{k},\frac{1}{k},...,\frac{1}{k}\right\}\enspace , \nonumber
\end{align}
and the linearized BP equations are then
\begin{eqnarray}\label{eq:lbp}
\Delta^{i(t)\to j(t)}&=&\sum_{\ell(t)\in\partial i(t)\backslash\, j(t)}U\Delta^{\ell(t)\to i(t)}+ V\Delta^{i(t-1)\to i(t)}+ V\Delta^{i(t+1)\to i(t)}\nonumber\\
\Delta^{i(t)\to i(t\pm 1)}&=&\sum_{\ell(t)\in\partial i(t)}U\Delta^{\ell(t)\to i(t)}+ V\Delta^{i(t\mp 1)\to i(t)} \enspace ,
\end{eqnarray}
where $\partial i(t)$ means neighbors of $i(t)$, $U$ and $V$ denote derivatives evaluated at the factorized fixed point:
\begin{eqnarray}
U_{sr}=&\left . \frac{\partial \mu_s^{i(t)\to j(t)}}{\partial \mu_r^{\ell(t)\to i(t)}}\right |_{\frac{1}{k}}=\left . \frac{\partial \mu_s^{i(t)\to i(t\pm1)}}{\partial \mu_r^{\ell(t)\to i(t)}}\right |_{\frac{1}{k}}
\qquad
V_{sr}=\left . \frac{\partial \mu_s^{i(t)\to j(t)}}{\partial \mu_r^{i(t\pm 1)\to i(t)}}\right |_{\frac{1}{k}}=\left . \frac{\partial \mu_s^{i(t)\to i(t\pm1)}}{\partial \mu_r^{i(t\mp 1)\to i(t)}}\right |_{\frac{1}{k}} \enspace .
\label{eq:5}
\end{eqnarray}
Solving Eq.~\eqref{eq:lbp} amounts to finding eigenvectors of the Jacobian matrix $B$ composed of derivatives of the BP messages. However, the size of the matrix $B$ is
$\left(cn\times T+2n\times (T-1)\right )^2$, which is relatively large for an eigenvector problem. Using the non-backtracking matrix approach~\cite{Krzakala2013}, we convert this problem into a smaller eigenvector problem of size $4nT\times 4nT$ by defining
\begin{align}
\label{eq:B}
B' = \begin{pmatrix}
\lambda \mathbb{A}^{\mathrm{spatial}} & -\lambda \mathbb{I} & \lambda \mathbb{A}^{\mathrm{spatial}} & 0 \\
\lambda (\mathbb{D}^{\mathrm{spatial}} - \mathbb{I}) & 0 & \lambda \mathbb{D}^{\mathrm{spatial}} & 0 \\
\eta \mathbb{A}^{\mathrm{temp}} & 0 & \eta \mathbb{A}^{\mathrm{temp}} & -\eta \mathbb{I} \\
\eta \mathbb{D}^{\mathrm{temp}} & 0 & \eta (\mathbb{D}^{\mathrm{temp}} - \mathbb{I}) & 0
\end{pmatrix} \enspace ,
\end{align}
where $\mathbb{I}$ denotes the $nT$-dimensional identity matrix; $\mathbb{A}^{\mathrm{temp}}$ is the adjacency matrix of temporal edges with $\mathbb{A}^{\mathrm{temp}}_{(u,t),(v,t')} = \delta_{uv} ( \delta_{t,t'+1} + \delta_{t,t'-1} ) \,$; $\mathbb{D}^{\mathrm{temp}}$ is the diagonal matrix of temporal degrees with $\mathbb{D}^{\mathrm{temp}}_{(u,t),(u,t)} = 2$ if $0 < t < T$, and $1$ if $t=0$ or $t=T$; $\mathbb{A}^{\mathrm{spatial}}$ is the $nT$-dimensional matrix consisting of all the spatial edges, i.e., $\mathbb{A}^{\mathrm{spatial}} = \bigoplus_t A^{(t)}$ meaning $A_{(u,t),(v,t')} = \delta_{tt'} A^{(t)}_{uv} \, $
; $\mathbb{D}^{\mathrm{spatial}} = \bigoplus_t D^{(t)}$ is the diagonal matrix of spatial degrees where $D_{(u,t),(u,t)} = D^{(t)}_{uu}$.
We now obtain a spectral clustering algorithm using $B'$ in the following way:\ given a spatiotemporal graph, we construct matrix $B'$, then take vectors composed of first $n$ entries of eigenvectors associated with the largest (absolute) eigenvalues, and finally perform $k$-means clustering on matrix composed of the vectors. This yields a partition of the nodes; if desired number of clusters is two, then we simply use the sign of entries of the vector to separate nodes into two communities.
\begin{figure}[t!]
\centering
\hspace{-8mm}
\begin{tabular}{cc}
\hspace{-9mm} \includegraphics[width=0.48\textwidth]{Fig_2a.pdf} &
\hspace{-3mm} \includegraphics[width=0.60\textwidth]{Fig_2b.pdf}
\end{tabular}
\vspace{-7mm}
\caption{(\textit{left}) Spectrum (in the complex plane) of matrix $B'$ for a network generated by our model with $n=300, c=3, k=2$ groups and $(\epsilon,\eta)=(0.05,0.5)$. The complex eigenvalues are circumscribed by the circle. (\textit{right}) Overlap as a function of $\epsilon$ for different values of $\eta$ (given in the legend). The detectability thresholds for each choice of $\eta$, according to Eq.~\eqref{eq:threshold}, are shown as vertical lines in lower panel, and the hatched area shows the region of detectability for static networks~\cite{decelle2011asymptotic}. Each data point is the average of 100 instances of dynamic networks from our model, with $n=512$, $T=40$, and $k=2$ groups, with average degree $c=16$.}
\label{fig:2_total}
\end{figure}
From the principle of linearization, we know that real eigenvalues of the non-backtracking matrix $B'$ describe stability properties of fixed points of the BP equations, i.e., if there is a real-valued eigenvalue larger than unity, it represents a stable fixed point in the equations. Moreover, if the BP equations have a stable fixed point, then $B'$ should have a real-eigenvalue that is larger than unity, denoting a partition of the nodes that correlates with the latent community labels. Thus, our spectral clustering algorithm should work as long as BP works, implying that it also works all the way down to the detectability transition in sparse networks.
In Fig.~\ref{fig:2_total} (left) we show the spectrum of $B'$ in the complex plane for a network in the detectable regime, generated by the model. As with existing non-backtracking approaches~\cite{Krzakala2013}, most of the eigenvalues are confined to a disk, while several real eigenvalues fall outside this disk. In this example, entries of the eigenvector associated with the largest real eigenvalue have the same sign, hence the leading or ``ferromagnetic'' eigenvector does not yield information about the latent community structure. In practice, we can perform regularizations to push such ferromagnetic eigenvectors back into bulk, thereby lifting the eigenvectors correlated with the latent community structure to the top positions. Eigenvectors associated with other real eigenvalues outside the bulk are correlated with the latent community structure. In this case, because we have two groups, we obtain the inferred partition by using the sign of entries of second real eigenvector $v_2$.
\section{Numerical verification}
\label{sec:experiments}
To verify our claims of the detectability transition in dynamic networks, and the accuracy of our algorithms, we conduct the following numerical experiment. Using our generative model of dynamic networks with community structure, we generate a number of dynamic networks for various choices of $(\epsilon,\eta)$. When $\epsilon=c_\mathrm{out}/c_\mathrm{in}=0$, communities are maximally strong, with every edge being located within a community, while at $\epsilon=1$, we have Erd\H{o}s-R\'enyi random graphs with no community structure. We then use our BP or spectral algorithm to infer the group assignments, assuming within each sequence that parameters $\{\eta,\epsilon,c\}$ are known. For each choice of $(\epsilon,\eta)$, we average our results over 100 dynamic networks with $T=40$ graphs and $n=512$ nodes (for $20,480$ nodes total), with an average degree $c=16$, divided into $k=2$ latent communities.
We measure the accuracy of the inferred community labels by the \textit{overlap} between the latent partition $g^*$ and the inferred one $\hat{g}$. This is the fraction of nodes labeled correctly, maximized over all $k!$ permutations of the groups, normalized so that it is $1$ if $\hat{g} = g^*$ and $0$ if $\hat{g}$ is uniformly random. In Fig.~\ref{fig:2_total} (right) we show the overlap obtained by BP for dynamic networks as a function of $\epsilon$ for several choices of $\eta$. The detectability threshold for each $\eta$, from~\eqref{eq:threshold} is shown as vertical lines in the lower panel. When $\eta=0$, we recover the static detectability threshold given by Eq.~\eqref{eq:static:thre}. As we increase $\eta$, the phase transition occurs at increasing values of $\epsilon$, as predicted, with the largest increase occurring when $\eta=1$.
Similar results are obtained for other choices of $n$ and $T$, with better agreement for larger networks. The slight deviation between numerical and analytic transition points observable in Fig.~\ref{fig:2_total} right is a finite-size effect, which we numerically estimated to decrease like $O(\sqrt{nT})$.
Figure~\ref{fig:hmap_final_phase} show the overlap throughout the $(\epsilon, \eta)$-plane, using both BP and spectral algorithms, along with the line of the threshold given by Eq.~\eqref{eq:threshold}. Notably, both algorithms perform similarly:\ they have large overlap with small $\epsilon$, indicating that the learned partition is highly correlated with the latent community structure. As $\epsilon$ increases (weaker community structure), both algorithms encounter a second-order phase transition in which the overlap decreases from a finite value to zero. Separate numerical experiments indicate that the convergence time of BP diverges in the vicinity of the phase transition, which agrees with past work on the detectability threshold in static networks~\cite{decelle2011asymptotic}. We also find that at each point in ($\epsilon, \eta$)-plane, the accuracy of BP is always larger than that of the spectral algorithm, especially away from the transition, reflecting the optimality of our BP algorithm.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\hspace{-6mm} \includegraphics[width=0.60\textwidth]{Fig_3a.pdf} &
\hspace{-20mm} \includegraphics[width=0.60\textwidth]{Fig_3b.pdf}
\end{tabular}
\end{center}
\vspace{-3mm}
\caption{Heat maps showing the numerically estimated overlap for (\textit{left}) belief propagation and (\textit{right}) spectral algorithms. The detectability threshold from Eq.~\eqref{eq:threshold} is shown as a solid line. Each point shows the average over 100 instances of dynamic networks drawn from our model with $n=512,$ $T=40$, $k=2$ groups, and average degree $c=16$.
\label{fig:hmap_final_phase}
}
\end{figure}
\section{Conclusions}
\label{sec:dis}
We have derived a mathematically precise and general limit to the detectability of communities in dynamic networks. This threshold assumes a probabilistic model of community structure that is a special case of several previously developed methods to detect dynamic communities: specifically, where nodes may change their community membership over time, but where edges are generated independently at each time step. We also gave two efficient algorithms for learning latent community structure that are optimal in the sense that they succeed all the way down to the detectability threshold in dynamic networks.
A simple extension of our algorithm is to apply our BP equations to a dense network consisting of all spatial edges from all graphs projected to the time $t$, handling the message passing over time steps by using a damping factor $\tau^{|t-t'|}$. This approach extends our analysis to networks that evolve in continuous time rather than in discrete time steps.
For larger numbers of groups, such as $k > 4$, it has been conjectured~\cite{decelle2011asymptotic} that there is a ``hard but detectable'' regime where the factorized fixed point described in Section~\ref{sec:spectral} is locally stable, but where one or more accurate fixed points exist as well. In such a regime, community detection is information-theoretically possible, but we believe that it takes exponential time (though see~\cite{kanade-mossel-schramm} for the case where the number of groups grows with $n$). We propose this as a direction for further work.
Other directions for future work include handling cases where the community interaction matrix $p$ may also change over time (a situation similar to change-point detection in networks~\cite{peel2015detecting}), where edges are not generated independently at each time step, or where networks have edge weights~\cite{aicher2014wsbm} or node annotations.
\begin{acknowledgements}
The authors thank Elchanan Mossel and Andrey Lokhov for helpful conversations. Financial support for this research was provided in part by Grant No.\ IIS-1452718 (AG, AC) from the National Science Foundation, Grant \#FA9550-12-1-0432 from the U.S.\ Air Force Office of Scientific Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA) (LP), and the John Templeton Foundation (PZ, CM). Author order is joint first-authorship for AG and PZ, with the remaining authors appearing alphabetically.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,758 |
{"url":"https:\/\/bramblexu.org\/2020\/01\/27\/scientific-writing-and-latex-tips\/","text":"# Scientific Writing\n\n## Grammar and Style\n\n\u2022 Training set, validation set, test set.\n\n\u2022 Use of \u201cSOTA\u201d. \u201cState of the art\u201d as a noun is written without hyphens, as in \u201cOur algorithm represents the state of the art\u201d. As an adjective, \u201cstart-of-the-art\u201d is hyphenated, as in, \u201cThese are state-of-the-art results\u201d. Also, don\u2019t forget the determiner in the nominal form, i.e., the state of the art.\n\n\u2022 Related work. \u201cRelated work\u201d, not \u201crelated works\u201d. \u201cWork\u201d is a collective noun, like \u201cstaff\u201d. You don\u2019t write \u201cresearch staffs\u201d.\n\n\u2022 That vs Which. The meaning of the sentence is not properly conveyed if the that clause is not included. If you remove the which clause from the sentences, the meaning of each sentence is still conveyed. See detail in here\n\n\u2022 Resolving ambiguity using \u2018and\u2019:\n\n\u2022 To my parents, Ayn Rand and God. This is ambiguity because \u201cAyn Rand and God\u201d can be read as in apposition to my parents. A comma before and removes the ambiguity. To my parents, Ayn Rand, and God. See detail in Serial comma\n\n# LaTeX Tips\n\n## Appearance\n\n### Author\n\n\\author{\n\\begin{tabular}{c@{\\ \\ \\ }cc}\nXu Liang & Yasufumi Taniguchi & Hiroki Nakayama \\\\[3pt]\nTIS Inc. & TIS Inc. & TIS Inc. \\\\[3pt]\n% \\multicolumn{3}{c}{TIS Inc. TIS Inc.} \\\\\n\\vspace{-4ex}\n\\end{tabular}}\n\\date{\\texttt{\\{ryo.sho, taniguchi.yasufumi, nakayama.hiroki\\}@tis.co.jp}}\n\n### Math Function\n\n\u2022 Conditional Probability : $P(x \\mid y)$, ues P(x \\mid y) , instead of P(x | y)\n\u2022 Adding equation number only if we want to cite it in other place.\n# turn on equation numbering\n\\begin{equation} \\label{eu_eqn}\ncoverage = \\frac{\\sum \\delta (e_i, L)}{\\sum_{i=0}^{n}e_{i}},\n\\end{equation}\n\n# turn off equation numbering\n\\begin{equation*} \\label{eu_eqn}\ncoverage = \\frac{\\sum \\delta (e_i, L)}{\\sum_{i=0}^{n}e_{i}},\n\\end{equation*}\n\n### Code\n\nUsing verbatim or listing:\n\n\\begin{verbatim}\nyour\ncode\nexample\n\\end{verbatim}\n\n### Footnote & URL\n\n\u2022 Footnotes go after the punctuation, as in End of sentence.\\footnote{...}. Note, there is no space between punctuation and the footnote. And it also go after the closing parenthesis. See detail here.\n\u2022 Use \\url for URL representation.\n\\usepackage{hyperref}\n\n\\noindent \\textbf{Data}~In our experiments, we use two Japanese annotated corpora, the Balanced Corpus of Contemporary Written Japanese~(BCCWJ)\\footnote{\\url{https:\/\/pj.ninjal.ac.jp\/corpus_center\/bccwj\/}}, and Mainichi Newspaper Corpus.\\footnote{\\url{http:\/\/www.nichigai.co.jp\/sales\/mainichi\/mainichi-data.html}} According to the entity annotation scheme,\\footnote{\\url{https:\/\/nlp.cs.nyu.edu\/ene\/ene_j_20160801\/Japanese_7_1_2_160917.htm}} these datasets contains multiple entity types. But we only extract the samples that contain the Company\" type. There are total 4,391 sentences from two datasets~(see Table~\\ref{t:dataset}). \n\n### Quotation\n\nDouble quotation in latex:\n\nThe Company'' type is ...\n\n### Table\n\nUse LaTeX Tables Generator to generate the table, and insert it in the between \\begin{tabular} and \\end{tabular} .\n\nOne column table begin with table , and resizebox(0.45\\textwideth) could resize the table size, and use the \\centering to center the words:\n\n\\begin{table}[t]\n\\centering\n\\resizebox{0.45\\textwidth}{!}{\n\\begin{tabular}\n# insert the tabular here\n\\end{tabular}\n}\n\\caption{Lexicons statistics}\n\\label{t:lexicon}\n\\end{table}\n\nTwo column table begin with table* . Do not use \\vspace{-0.65cm} to adjust the space, avoiding dest reject. If the space is not enough, you could read your paper few times and delete some reduent content.\n\n\\begin{table*}[]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|}\n# insert the tabular here\n\\end{tabular}\n\\caption{Intrinsic evaluation ...}\n\\label{t:coverage}\n% \\vspace{-0.65cm}\n\\end{table*}\n\nA desk reject means that the program chairs (or editors) reject a paper without consulting the reviewers. This is done for papers that fail to meet the submission requirements, and which hence cannot be accepted. Filtering out desk rejects in advance is common practice for both conferences and journals.\n\nFix table position, let the table under text.\n\n\\usepackage{placeins}\n\n\\FloatBarrier\n\\begin{table}[h]\n\\resizebox{0.45\\textwidth}{!}{\n\\centering\n\\begin{tabular}\n# insert the tabular here\n\\end{tabular}\n}\n% \\caption{Alias generation process}\n% \\label{t:alias}\n% \\vspace{-0.65cm}\n\\end{table}\n\\FloatBarrier\n\nTable alignment: the text in the left 1-th column should be left aligned, and the numbers in the right column should be right aligned:\n\n### Refer to a figure or table\n\nThere is no need to add \u2018the\u2019 before Table~\\ref{t:coverage}.\n\nTable~\\ref{t:coverage} list the coverage scores for different lexicons and datasets. We can see the JCL lexicon cover most company names~(0.5082 in Mainichi and 0.5407 in BCCWJ) than other single lexicon. As for the multiple lexicons, JCL also boost the coverage score of IPAdic-NEologd~(from 0.5291 to 0.6923 in Mainichi, from 0.4593 to 0.6522 in BCCWJ).\n\n### Citation\n\n\u2022 Cite a reference as Lin et al.~\\cite{Lin}. Note ~ prevents ugly breaks.\n\n\u76ee\u5f55","date":"2020-04-05 13:47:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6976193785667419, \"perplexity\": 8836.276337357418}, \"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-2020-16\/segments\/1585371604800.52\/warc\/CC-MAIN-20200405115129-20200405145629-00499.warc.gz\"}"} | null | null |
Analyzing Mustapha Heron and Kostas Antetokounmpo
I decided to tune in to watch Auburn and Dayton because I wanted to get a glimpse at seeing Dayton's freshman big man, Kostas Antetokounmpo play, as he is the younger brother of Milwaukee Bucks' superstar forward, Giannis Antetokounmpo. But in the actual game itself, it was not his play that impressed me, but rather the performances of other players that were on Auburn's team. Mustapha Heron had a really solid game, as did others such as Chuma Okeke and Anfernee McLemore. Here are my thoughts on how various prospects fared on this day.
Auburn's 6-5 sophomore guard, Mustapha Heron played very well, as he excelled as a shooter, scorer, and rebounder when he was on the floor to help his team to get a 73-60 win over Dayton. He did a good job of getting scores off the dribble. Heron is a patient player that possesses good strength, quickness, and the shooting touch to garner numerous scores and free throws on aggressive drives to the basket.
Plus, he really excelled at knocking down jumpers. Though Heron has struggled to make them so far this season, this may have been his best game of the year to date. Heron showed a knack for making three-point shots, and he can knock down contested pull-up threes effortlessly, as he can also use a step back move. He struggled more to make mid-range jumpers, but generally Heron did a great job of making his jump shots.
He primarily looked to score the basketball, but he did handle the ball very well, as he can deftly maneuver around to get shots up off the dribble from anywhere on the court. Heron also rebounded well, as he would leap up to get numerous defensive boards. He also displayed decent defensive skills. Heron showed a good ability to contest shots to force misses, though sometimes he would be too aggressive in trying to make plays on the ball, which would lead him to commit some fouls. In general, Mustapha Heron had a solid game in his team's win over Dayton. He is an athletic swingman that can shoot and score the basketball, and if he continues to play well, he could perhaps end up being a first round pick in the upcoming draft.
I tuned in to this game to watch Dayton's 6-10 freshman forward, Kostas Antetokounmpo play, and at first, it seemed that he would be on his way to have a good game. On the first play of the game, he won the tip in a convincing fashion, and he showed a solid first step off the dribble to get past his man, as he appeared to have scored on a quick straight-line drive to the hoop. However, that play was later waved off, as the referees ended up calling a charging foul on him, which negated the would have been score, resulting in Kostas committing a turnover.
He then quickly picked another foul, as he committed a reach-in foul when defending the post, as he was sent to the bench early in the first half. He came in later with a couple of minutes remaining in the first half, but it wasn't until the second half where he would make some positive contributions on the floor, though in relatively small doses.
Antetokounmpo showed a strong ability to finish plays around the basket. On one play, he was open around the basket, and he caught the ball to throw down a thunderous dunk to help his team. He also showed an adequate ability to score off the dribble, and in the second half he managed to draw free throws on a quick drive to the basket. He will still need to improve upon his dribble moves, though. He will also need to improve his ability to shoot the basketball, as he threw up an airball to miss on a mid-range shot completely, and he continually would pass up three-point shot attempts despite being left wide open.
Antetokounmpo showed decent playmaking skills. He is an unselfish player that will look to make the extra pass in order to find the open man. The turnover he committed was questionable, as the defender guarding him that led to his charging foul appeared to have flopped when guarding the drive. However, he showed a poor ability to rebound the basketball, as he literally came away with no boards in this game against Auburn. He did not show great instincts to try to haul them in, and he would also tend not to block out, though one play was late in the half, as he already had two fouls and he didn't want to risk picking up a third foul. As a result of that, his team did end up allowing a put back dunk towards the end of the first half.
He also showcased decent defensive skills. Antetokounmpo showed a gifted ability to block shots, as he is a very athletic big man that possesses a solid blend of size, leaping ability, and timing to do so. He also managed to get his hands up to deflect an entry pass out of bounds. Also, he was able to stay in front of his man on drives and contest jumpers to force misses.
However, he had trouble defending in the post, as he would commit bad reach-in fouls, and he also lacks the strength to adequately guard stronger opponents on the block, but sometimes, he would not rotate in time to timely contest shots off drives, and he also gave up too much space once to allow his opponent to score on a mid-range shot.
In this game, Kostas Antetokounmpo could be a player to keep an eye on, as he has great size, athleticism, and he may have a variety of skills that could prove to be useful if he continues to work on his overall game. However, he still may be very unpolished as an overall player, as he is still learning the game, and he will need to improve his perimeter skills, as well as find more ways to consistently stay on the floor and avoid foul trouble. He will likely need to stay in school to hone his game, but if he decides to leave school to enter the 2018 NBA Draft, it's conceivable that an NBA team could take a flyer on him in the mid to late second round as a developmental project that they could invest in for the long-term.
Auburn's 6-8 freshman hybrid forward, Chuma Okeke had a solid game, as he played especially well in the first half. He is an athletic basketball player that possesses solid scoring skills, and he shot the ball really well from downtown. He was able to make numerous three-point shots from beyond the arc, including one where he used a pump fake and a step back move. Okeke can make shots with hands in his face, as well as pull-up threes off of step back moves.
He also is a good inside scorer as well, as he can score on hustle plays as well as when he is on the block. His first basket was when he ran off of an off-ball screen to cut to the hoop,, as he was able to catch a lob pass to score on a nifty, alley-oop dunk. Plus, he also was able to score on a drop step move in the post. He also showed a willingness to pass the ball, and he can spot open teammates to set up scoring chances, but sometimes he would get a ball poked away or one of his passes deflected.
Okeke did a very good job of rebounding the basketball, as he would actively crash the glass to haul in numerous boards on both ends of the floor, and he also managed to draw free throws off of a put back once. Defensively, he looked to be average. He was able to deflect a pass, as he can play passing lanes. But on one play, he overplayed a shooter and bit on a pump fake, which led him to get beat on a drive to allow a score.
In general, Chuma Okeke played very well, especially in the first half, as his strong play in this game helped his team get a road win over Dayton. He may not be receiving much draft buzz right now, but he may possess an intriguing skill set, as he could end up being a potential first round pick for an NBA team in the future.
Dayton's 6-5 senior point guard, Darrell Davis had a fairly good game in his team's loss to Auburn. He played quite well in the first half, and he is a smart, skilled player that can shoot and score the basketball and also effectively run the team at the point. Davis showed a good ability to knock down threes from beyond the arc. He especially excelled at making pull-up, three-point shots, as he even made one that was tightly contested, but he also struggled to make spot-up threes, as some of them were closely contested, and other times he would rush those shots.
He also was able to score on some hustle plays. Davis managed to score on a put back in traffic after getting his own miss, and he also was able to score in transition. However, sometimes Davis would try to force the issue, as he would end up getting his shots blocked or missing shots in traffic, and he had more trouble trying to score on drives when he would attempt to go all the way to the hoop.
Plus, he also showed off fairly good playmaking skills. Davis was at his best when he would play with patience, as there were times where he would demonstrate solid court vision. He tends to throw solid passes off the dribble, as he can hit cutters and kick the ball out to shooters. He also can find teammates on the move. However, sometimes he can be turnover prone due to carelessness. He committed a charging foul in the post once, and he also had one of his passes intercepted. Another time, he took his eye off the ball, and he ended up dropping a catchable pass out of bounds.
He also did a good job of getting rebounds when he was on the floor. Davis has good instincts, and he was able to haul in numerous defensive boards in this game, but there was one play where he didn't block out a bigger player, as his teammate ended up committing a foul on a put back. Also, he displayed decent defensive skills. Davis was able to leap up to break up a lob pass, and he showed an adequate ability to challenge jump shots to force misses. But sometimes, he would get beat on drives, which would him or his teammates to commit fouls.
In general, Darrell Davis had a fairly good game, as he is a skilled basketball player that can contribute in a variety of ways on offense, and he has the ideal size to play either backcourt position when he is on the floor. However, he only possesses average athletic gifts for the next level, and he may not project to be an especially dynamic playmaker in the NBA, but with his skill set and with the intangibles he brings to the table, Davis could end up being a solid addition to an NBA team's summer league roster.
Auburn's 6-7 sophomore forward, Anfernee McLemore played very well, as he made a lot of solid, energetic plays on both ends of the court to help his team to defeat the Dayton Flyers. McLemore excels at scoring on hustle plays, as he managed to score on a thunderous put back dunk late in the first half, and he also scored on a floater off of a cut. Plus, he also can score in the post, and he does a good job of running off of screens to get open to catch the basketball.
Also, he rebounded and defended very well when he was on the floor. He grabbed many rebounds, as he was basically like a chairman of the boards. McLemore also did a great job of protecting the paint as he blocked many shots to do so, and he even chased down an opponent to make a nice chase down block from behind to prevent an opponent from scoring on a drive in transition. McLemore has very good athleticism, leaping ability, and timing, and he can swat many shots around the basket to force misses, but every once in a while, he may commit fouls due to over-aggressiveness, as he may try to block every shot.
McLemore is a very athletic combination forward that has excelled as a solid role player for his team, but it may be unclear if his skills will translate to the NBA. He has the game of a high motor, energetic, shot blocking, rim protector, but he is only a 6-7 forward, and he does not possess a whole lot of perimeter skills on offense.
Right now, he may project to be a tweener though, which could prevent him from getting picked high if he were to declare for the NBA draft. Currently, he may possess a similar game to that of former NBA player, Dominic McGuire. In the right scenario, McLemore could perhaps develop into an effective role player at the next level if an NBA team decides to develop his skill set for the long haul. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,689 |
import platform
is_pypy = platform.python_implementation() == 'PyPy'
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,001 |
module Vault
# `Vault::Test` provides support for writing automated tests.
module Test
end
end
require "vault-test-tools/version"
require 'rack/test'
# Test case mixins.
require "vault-test-tools/environment_helpers"
require "vault-test-tools/html_helpers"
require "vault-test-tools/logging_helpers"
require "vault-test-tools/spec_helpers"
# Test case base classes.
require "vault-test-tools/test_case"
require "vault-test-tools/spec"
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,224 |
The longest floating structure in history is about to hit the ocean. It might fix a big problem.
The design might seem simple, but it's definitely not.
In 2016, "the longest floating structure in world history" will be placed in the ocean.
Don't worry — it's not another super yacht or party barge or some other contraption that will further pollute the ocean. Nope, this one is a good thing.
It's called The Ocean Clean-up, and it's a 1.2-mile-long system designed to collect and remove plastic from the ocean.
For two years, it will hang out in the ocean hopefully to begin undoing what we've done for decades: polluted the heck out of the water with plastic trash.
It's basically a stationary array of barriers that uses the ocean's natural currents to collect the plastic at a central location.
When I first wrote about The Ocean Clean-up a year ago, I thought it was something that moved through the ocean, collecting trash as it went. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,960 |
Here we compare ADT monitored security from Protect Your Home and Cox home security's basic packages. We hope this side-by-side comparison makes it easier for you to make a decision on which company better fits your home security needs. In addition, our expert opinion might help you get clear about your home security decision.
Source data: https://www.cox.com/residential/homelife/pricing.html. Last updated Jan 2017.
"Cox offers home security plus home automation for a $49.99 activation fee and $50 installation fee plus $39.99 ($29.99 per month if you bundle with other Cox services) per month for monitoring services. Their alarm package only includes two door and window sensors and no remote control. This is fine for an apartment or small condo, but it's not adequate coverage for most homes.
Another thing you need to realize is that even though home automation is being included as part of the monthly monitoring service, the up-front investment to automate your home is costly. You'll need a new thermostat ($300), X10 light modules for any lights you want to control ($100 each), video cameras ($250 each), plus the cost of an electrician to install it all.
Disclosure: Above opinion is my own opinion and I do not receive compensation to provide my opinion. My opinion does not reflect the opinion of Protect Your Home. Last updated on Jan 2017.
Ready To Get Started With ADT Monitored Security?
If our review of Cox has convinced you that they are not the right choice for your home's security needs, you can speak with an ADT Authorized Premier Provider representative by calling 1-866-946-8419. Be sure to ask about next day installation!
To compare ADT monitored security from Protect Your Home to other home security companies, simply click on the company below you would like to compare. You'll be taken to a page that includes a side-by-side comparison for your convenience. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,237 |
\section{Introduction}
\label{sec:intro}
The emergence of the internet and the global success of social media has fundamentally changed how people search for information, how they are connected to each other, and how they communicate with each other \cite{Vorderer2015d}. Social media platforms like Facebook, Instagram, or Twitter as well as instant messengers like WhatsApp have become an established part of the modern media diet~\cite{Newman2017}. In 2018, a majority of Americans used Facebook and nearly a quarter was active on WhatsApp \cite{Smith2018}. In other countries, the proportion of WhatsApp users is even higher: about 70\% of the Germans use WhatsApp - often on a daily basis~\cite{Whatsbroadcast.com2018}. Social Media and Instant Messengers offer enhanced possibilities for interpersonal communication largely without geographical or temporal boundaries. Additionally, in some regions, social media is increasingly used as central communication and information channel, e.g., WhatsApp in India~\cite{StatistaIndia2016,Raina2018}.
Not all communication partners in online-interactions are humans.
Fully or semi-automatized user-accounts, so-called \emph{Social bots}, increasingly participate in online interactions.
In contrast to other automated agents (such as web-crawlers or service bots), social bots are designed for one- or many-sided communication and the imitation of human online-behavior~\cite{Grimme2017,Frischlich2018}. The spectrum of (assumed or observed) types of social bots ranges from very simple bots that automate single elements of the communication process (e.g. liking or sharing), over partially human-steered accounts with automated elements (so-called hybrid bots, or "Cyborgs", see~\cite{Chu2010}) to autonomously acting agents~\cite{Grimme2017} equipped with artificial intelligence and learning skills such as Microsofts' Zo (https://www.zo.ai/). Currently, social bots and their influence are especially discussed in the context of manipulation and disinformation~\cite{KollanyiUSEl2016,Ferrara2016,bessi2016social}. However, the detection of social bots remains a challenge~\cite{Grimme2018}, wherefore the actual number of social bots and also details of realization are unclear. Varol and colleagues~\cite{Varol2017} estimated that in 2017, a fraction of 9-15\% of the active twitter accounts were social bots, while platforms themselves report on millions of accounts~\cite{Roth2018}
It is clear, that Social bots need a technical online infrastructure, whic
which in general can be broadly understood as the combination of (a) a user-account on a social network site, a micro-blogging service, or an instant messenger; and (b) the technical preconditions for partial automation of the accounts-behavior via the accordant platform's API or proprietary mechanisms to interact with the website or app front-end of the account (this is virtually equivalent to remote controlling an app or browser). Additionally, the technical perspective includes the algorithmic realization of the account's behavior. While a basic social bot may only perform simple amplification activities (liking or sharing), more advanced bots may possibly replicate human behavior, post messages, or even directly interact with other accounts (humans or bots).
Apart from multiple often theoretical or anecdotal taxonomies, only little is known about the relative availability of these different types of social bots. From the technical perspective, the term "availability" implies three consecutive research questions to be posed:
\begin{enumerate}
\item \emph{What amount of ready-to-use code is publicly available to realize social bots?} This question addresses the general and free availability of source code in form of automation, access, and remote control frameworks, or ready-to-use code blocks to build social bots.
\item \emph{What role does machine learning and artificial intelligence play for current off-the-shelf bot development?}. Considering the multiple taxonomies for social bots as well as public discussion on social bots, both (at least implicitly) assume the existence of what is often called "intelligent" bots. This question investigates whether a corresponding development is identifiable in open bot code.
\item \emph{Consequently, what are the costs for creating social bots of different complexity levels (simple up to artificially intelligent)?} Depending on the availability of ready-to-use code for social bot development and based on the findings regarding available ingredients for constructing simple up to "intelligent" bots, we can roughly infer the effort, which is necessary to construct different types of social bots.
\end{enumerate}
This work strives to answer the questions by exploratory knowledge discovery using data from open-access social bots development projects, which is shared on different online-repositories, as an estimator. Online repository and collaboration infrastructures such as the
platform \texttt{Github}\footnote{\url{https://github.com/}} have become a central hub of current software development processes~\cite{Zagalsky2015} and thus provide meaningful insights into the available knowledge in a certain programming area.
Based on the insights gained from the worldwide available code for social bots, we propose a set of analytic methods that provides a broad picture to which extent social bots can be constructed from freely available code building blocks, frameworks, and technology snippets. This enables us to judge on the common stage of development of social bots "in the developer market" and can provide - on the long term - a way to detect new developments, threats, or trends in social bot evolution.
The paper is structured as follows: Section~\ref{sec:related_work} provides a literature overview on related work and the state of knowledge about social bot technology. Section~\ref{sec:data_sources} details the data source selection, relevant social media platforms, and data acquisition process. The analysis starts in Section~\ref{sec:descr_analysis} with a description of the acquired data and is continued in Section~\ref{sec:qual_analysis} with the data exploration and qualitative analysis of the social bot code descriptions using topic modelling via Latent Dirichlet Allocation (LDA)\cite{lda} and an innovative approach of stream clustering for texts~\cite{ER17Twitch}. Finally, in Section~\ref{sec:discussion}, we discuss our results and conclude our work.
\newpage
\section{Related Work}
\label{sec:related_work}
So far, research on social bots mainly addressed specific (sometimes learning- and thus data-based) approaches for the identification and observation of automated accounts in online media~\cite{Paradise2014,Cao2014,clark2016a,Ferrara2016,Varol2017,GG2018,Cornelissen2018ANT} or, discussed the influence of malicious social bots on the public debate~\cite{Fredheim2013,AYM2015,bessi2016social,Marechal2016} for instance via the distribution of hate or fake news~\cite{Frischlich2018,Shin2018,Vargo2018}.
Based on observations, detection of suspicious and clearly automated accounts, practice reports, and probably biased by public discussion, several social bot classifications of taxonomies have been proposed. As an early and rough classification Wooley~\cite{Woolley2016} distinguishes between classical bots for pure automation purposes and those who influence "public opinion or disrupt organizational communication". He defines such bots as "political bots". Hegelich~\cite{hegelich2016} follows a similar approach by defining two classes: "assistants" (also chat bots) and "social bots", where the latter are hidden actors in a political context. A more detailed categorisation is provided by Stieglitz and colleagues~\cite{Stieglitz2017DoSB}. They assume "imitation of human behaviour" and "intent" as two nominal scales, where the first discriminates low/none and high while the latter distinguishes malicious, neutral and benign. In this matrix, social bots imitate human behavior with malicious intent. Gorwa and Guilbeault~\cite{GG2018} address the "incredible breadth of terminology" used for (social) bots in literature. Focusing on the functions of bots, they suggest a more holistic typology: Crawlers and scrapers "working behind the scenes"; chat bots for human computer interaction using text; spam bots that spread advertisement, comments, or perform DDoS attacks in an automated and high frequency fashion; social bots, which the authors classify as web 2.0 versions of bots that often use APIs, however, are not clearly distinguishable in function from previously mentioned bots - except those, that they can act politically. In addition the authors also mention sockpuppets and trolls as well as cyborgs and hybrid accounts as partly automated profiles and thus special categories.
In an attempt to integrate the variety of perspectives, Grimme and colleagues~\cite{Grimme2017} propose a taxonomy that first distinguishes social and non-social bots (the latter corresponds to assistants mentioned by Hegelich~\cite{hegelich2016} or Gorwa and Guilbeault~\cite{GG2018}) and splits the class of social bots into three sub-classes: simple, advanced, and hypothetically "intelligent" social bots. For simple bots, the authors provide source code, which is able to perform very simple tasks like posting, sharing, or liking content. The second class is considered to imitate human behavior (in the sense of Stieglitz et al.~\cite{Stieglitz2017DoSB}) by acting in human speed, mimicking human inactivity, and simulating off-topic interests. Here, the authors report on experiments they performed with this kind of bots. Finally, they assume a possible third class of intelligent bots, that act on their own and virtually human-like, also in content production. However, as the authors emphasize, for this class no representative set of instances has yet been detected or published.
In fact, apart from an enormous set of online (gray) literature\footnote{e.g. BotWiki \url{https://botwiki.org/resources/twitterbots/} , Fredheim \url{http://quantifyingmemory.blogspot.co.uk/2013/06/putins-bots-part-one-bit-about-bots.html}, Grossmann \url{https://medium.freecodecamp.org/my-open-source-instagram-bot-got-me-2-500-real-followers-for-5-in-server-costs-e40491358340} }
or technology documentations\footnote{e.g. \url{https://github.com/eggheads/eggdrop/}, \url{https://dev.botframework.com/}}
, only very few scientific works back the postulated taxonomies or findings with explicit expertise on actual social bot code. Grimme et al.~\cite{Grimme2017,Grimme2018} are rare examples of this group. As such, the empirical evidence on available software to realize social bots as well as insight into the degree of development of such codes is very scarce. In a notable exception, in 2016, Kollanyi~\cite{Kollanyi2016} has examined the availability of open-source code for twitter bots on \texttt{Github}. He shows that the number of repositories providing twitter bot code has been steadily increasing since the launch of the platform, with the majority of repositories being provided by actors from the United State or Japan. In light of changing media system and the global success of platforms besides Twitter, it is, however, highly plausible that Twitter bots only form a small share of the overall social bots available. As such the study by Kollanyi --- albeit providing pioneering insights into the availability of open-source social bot code in general and their abilities more specifically --- might be limited when it comes to understanding the availability of social bots more generally.
The current study, however, builds on Kollanyi's approach and aims for a set of methodologically advanced tools for providing insights into (1) the availability of social bot code for the most successful social media platforms as well as instant messengers all over the world and (2) the skills these bots have implemented in their code related to the taxonomy of Grimme et al.~\cite{Grimme2017}.
\section{Data Sources}
\label{sec:data_sources}
\paragraph{Code Repositories:}
In the first step, a total of 54 active online code repositories were identified. As we were interested in the available knowledge about social bots, we focused on open platforms that allowed for collaboration and identification of specific codes via searchterms.
A total of eight repositories fulfilled these criteria and allowed for (a) version control via Git or Appache subversion (SVN); (b) collaboration between users; (c) public access (that is they were researchable in the clear web); and (d) searching for specific terms.
For the analyses, we used Alexa global usage statistics to identify the five most relevant repositories. The Alexa rank, is a metric, which can be taken to evaluate the importance of a website\footnote{see: Alexa Rank: https://www.alexa.com/}. The metric combines calculations of internal homepage traffic such as page callings, and their development over time. Websites are ranked by their importance, where an Alexa score of "1", means that the website is most important. The ranking is seen critically in practice, since it is prone to get manipulated e.g. by click spamming and further does not differentiate between specific services or website purposes~\cite{alexa}. Nevertheless, it can be used to compare website services within the same topics of interest. Table \ref{Table_Top5} shows the final selection of relevant code repositories.
\begin{table}\small
\setlength{\tabcolsep}{20pt}
\centering
\caption{Top 5 code repository hosting platforms}
\label{Table_Top5}
\begin{tabular}{@{}llr@{}}
\toprule
\textbf{Rank} & \textbf{Code Repository} & {\textbf{Alexa Rank (September 2018)}} \\ \midrule
1 & GitHub & 72 \\
2 & Sourceforge & 351 \\
3 & Bitbucket & 945 \\
4 & GitLab & 2,819 \\
5 & Launchpad & 7,529 \\
\bottomrule
\end{tabular}
\end{table}
\paragraph{Social Media Platforms:}
In order to describe the availability of different types of social bots, we focused on social media platforms, micro-blogging services, chat or Voice over IP-services, and instant messenger with the largest global reach. Reach was determined by a triangulation of (a) number of active users (see \cite{Statista2018}), (b) global traffic rank based on Alexa, and (c) downloads of the accordant application (e.g. via Google's Play Store or Apple's App Store). Where traffic information was missing (e.g. for the Instant Messengers) only applications with more than 500 Million downloads were included. As we focused on English-speaking repositories, the Chinese platforms were excluded from the key-word selection.
Based on the criteria, mentioned above, the following social media platforms were used within the data-acquisition process: \textit{Telegram, Twitter, Facebook, Reddit, Skype, Instagram, Youtube, Whatsapp, Linkedin, Tumblr, vKontakte, Snapchat and Pinterest}.
\paragraph{Data Acquisition:}
Since the collaboration platforms are differently structured, it was not feasible to establish a common and comparable procedure for searching for specific bot programs. The largest platform, \texttt{Github}, offers a detailed search engine where explicit search criteria can be applied on different repository fields. Similar to \texttt{Github}, \texttt{Gitlab} also offers an API. However, \texttt{Gitlab} is more limited when it comes to the specification of additional search criteria. \texttt{Bitbucket} only provides a single web-search interface without any documentation. Therefore it is infeasible to track which fields are used within the final search query. Given the diverse capabilities of the collaboration platforms, when it comes to the formulation of searchterm queries, we decided to dismiss any field restriction at all. Additionally we selected the searchterms as generic as possible. Concretely we combined the name of each Social Platform with the term \texttt{bot} via a logical AND operator. For \texttt{Github}, \texttt{Gitlab} and \texttt{Bitbucket} a unique crawler was programmed that automatically gathered the repositories information for all searchterm combinations. While \texttt{Github} and \texttt{Gitlab} explicitly provide an external API for searching, \texttt{Bitbucket} is not easily accessible. Therefore we utilized \texttt{Scrapy}, a python-based web scraping framework, for collecting the relevant information. The remaining platforms, Sourceforge and Launchpad were manually queried via the provided web interface because of the low number of matching repositories for those platforms. The scraped information was persisted within Elasticsearch, a document-based search engine which performs well on textual data.
To allow for time efficient crawling and avoid noise in the data set due to temporal developments during data collection, we specified the following limitations to our gathering process: First, we did not download the actual files (source code) of the repositories, since our analysis is mainly based on metadata. Secondly, we dismissed the history of individual commits (code contributions) on all repositories. Although these data may provide interesting insights, the amount of potential additional API requests would have been significantly increased. Instead, we limit our analysis to the first and last contribution. Due to the heterogeneous structure of the collaboration platforms, we defined a common intermediate schema for data representation. Although some platforms consist of fields (location attribute on \texttt{Github}) that are not present on other sources, we include these additional information sources in our analysis. This especially holds true for the \texttt{Github} platform which contains more than 90\% of all repositories.
\section{Descriptive Analysis}
\label{sec:descr_analysis}
In total the data of 40.301 different code-repositories was gathered between April 2008 and October 2018. The largest number of repositories was provided by \texttt{Github}(38.600), followed by \texttt{Gitlab}(1293) and \texttt{Bitbucket}(408).
Despite its high Alexa score, only 25 repositories were found on Sourceforge for all searchtearm combinations. Moreover, 10 of these repositories were maintained on \texttt{Github} in parallel. We explain this observation by the fact that Sourceforge is considered as one of the older collaboration platforms, with a lack of sophisticated functionality. Therefore, most developers probably decided to move to a different platform, which was able to fulfill their requirements. Also in 2013 and 2015 the platform was criticized for offering adware and bundled malware. As a result it was reported that users switched to other code-hosting platforms \cite{reutersgitlab}. For Launchpad, only 10 repositories were found. This is not a surprising result, since the platform is of small scale. In total the platform hosts only 13.000 repositories laying the focus on big, open source software projects such as MySQL, Inkscape or Unity.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figures/gitlab.pdf}
\caption{Number of new \texttt{gitlab} repositories over time}
\label{fig:gitlab}
\end{figure}
The largest competitors of \texttt{Github}, namely \texttt{Gitlab} and \texttt{Bitbucket}, provide only a small fraction of the total number of bot repositories (4\%) and are thus considered as niche platforms. Furthermore, we are able to observe the impact of Microsoft's recent announcement of aquiring the \texttt{Github} platform for 7.5 billion US dollars \cite{microsoftgit}. While the average number of new repositories on the \texttt{Gitlab} platform per month was 13.97 before the announcement, it drastically increased to 234 repositories in June and 117 in July 2018 (see Figure \ref{fig:gitlab}). As it was reported in various news reports, the announcement was negatively perceived by many open source developers, who publicly encouraged other developers to migrate to \texttt{Gitlab} \cite{reutersgitlab}. Obviously this affected the community of bot programmers as well.
\begin{figure}[h!]
\centering
\includegraphics[width = 0.7\textwidth]{figures/searchterm_distribution.pdf}
\caption{Searchterm distribution for all social-media platforms}
\label{fig:searchterm_dist}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/searchterm-development.pdf}
\caption{Number of Repositories for different platforms over time}
\label{fig:telegramA}
\end{figure}
Over all collaboration platforms, we observed a similar distribution regarding the number of repositories for a specific social-media platform (Figure \ref{fig:searchterm_dist}). Most of the identified programs were produced for Telegram, followed by Twitter, Facebook and Reddit. At first sight this is a surprising result since Telegram is not considered as one of the big social-media players and the platform only exists since 2013. A detailed inspection of the creation date for Telegram oriented repositories revealed that until 2015 the platform did not receive a lot of attention. This changed in June and July 2015, when a significant increase in the number of related projects can be observed. We can directly explain this sudden increase by the fact that on June 24, 2015, Telegram officially launched it's open bot platform, making it easy for programmers to create automated bot programs via an external API. Furthermore, the functionality of creating inline bots (bots that can be addressed in any chat) led to a second raise of newly created applications in January 2016. Hence the social platform itself seems to directly impact the community of social-bot code. Figure \ref{fig:telegramA} shows, amongst others, the number of newly created Telegram repositories over time.
In a second step we analyzed different lifespans of repositories. We define the lifespan of one single Repository as the time between the creation date and the last activity. Moreover an activity is characterized by any Repository interaction such as a new contribution, a fork or a newly assigned issue. We observe that more than 50\% of the crawled Repositories (18.000) have a lifespan of 0 days (Figure \ref{fig:telegramB}). This means that such Repositories were once created on a specific date and did not receive any further update after publication day. As indicated in \cite{Kollanyi2016}, some developers use the \texttt{Github} platform only as a medium for sharing their code rather than collaborating with other users.
\begin{figure}[h!]
\centering
\includegraphics[width=\textwidth]{figures/maintenance_with0_sqrt.pdf}
\caption{Lifespan of the crawled repositories}
\label{fig:telegramB}
\end{figure}
\paragraph{API Support and Programming languages}
Due to the heterogeneous structure and conflicting goals between different social media platforms, companies handle third party access to the service they provide in a different manner. Whilst some platforms actively encourage developers to create external applications by providing dedicated interfaces for accessing their data and functionality, some platforms do not offer such information. Within this work, we differentiate between four distinct classes of third party access.
\begin{itemize}
\item Social media platforms that are assigned to the \texttt{BotAPI} class are considered the most favorable ones for programmers. Such platforms do not only offer API's for third party institutions, but also dedicated services and functionality for bot programs. Within our study Telegram was the only platform that provided such sophisticated interface.
\item Platforms that offer an \texttt{API} to perform all the common tasks of the corresponding web-interface in an automated way. For most of the social platforms these tasks are following, creating comments and all kinds of flat social interactions (like/dislike).
\item Platforms that offer a \texttt{limited API} access. Although an interface for third parties exists, the range of functionality is limited. Platforms that are assigned to this class, for example only allow to access private user data.
\item Platforms that offer \texttt{no API} and no other means of interface for external parties.
\end{itemize}
Most of the social media platforms of interest offered some kind of API to third parties. Only Whatsapp and Snapchat do not provide any official API interface. Not surprising, those platforms are situated on the lower ranks regarding the number of repositories found for the specific search term. Most of the platforms with a higher rank do offer a more sophisticated API. In general we observed a positive rank correlation between the number of repositories found for a specific social platform and the corresponding level of API support ($\rho=0.78$, see Table \ref{tab:support-correlation1}). Overall, the \texttt{limited API} was the most prominent class to which social platforms are assigned. This can be explained by strict privacy policies of some bigger social-media platforms. Because of recent incidents, where private data was used for manipulation purposes (e.g. Cambridge Analytica), those companies were widely criticized by the public for providing unrestricted data-access. As a result, some platforms changed their policy and consequently limited their API functionality \cite{CamebridgeAPI}. Instagram, for example, only allows to access and analyze account related data and activities. In contrast to the past, external information like the access to the followers of arbitrary users is strictly prohibited, unless the target person explicitly gives the permission to acquire the desired information.
\begin{table}[tb]\small
\setlength{\tabcolsep}{9pt}
\begin{center}
\caption{API support for Social Platforms, sorted by number of repositories}
\label{tab:support-correlation1}
\begin{tabular}{@{}lcccc@{}}
\toprule
\textbf{Social Platform}& \texttt{no API}&\texttt{limited API} & \texttt{API} & \texttt{BotAPI} \\ \midrule
Telegram & & & & X \\
Twitter & & &X & \\
Facebook & & &X & \\
Reddit & & & X& \\
Skype & & &X & \\
Instagram & &X & & \\
Youtube & & X& & \\
Whatsapp & X & & & \\
Linkedin & & & X & \\
Tumblr & & X & & \\
vKontakte & & X & \\
Snapchat & X & & & \\
Pinterest & &X & & \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
Most of the social media platforms which provide a dedicated API, offer some additional interfaces to access their service. These interfaces can be accessed by specific programming languages. Within Figure \ref{fig:languages} we show an aggregated view of the repositories main programming languages for the top five social media platforms (in terms of repository count). The most common programming language over all platforms is Python. Interestingly JavaScript is also frequently utilized. While Facebook explicitly provides a Java Script Toolkit, this is not the case for the other platforms. In cases where the API is somehow restricted (e.g. companies privacy policy), programmers often directly access the web interface with JavaScript code to circumvent the official API.
\begin{figure}[b]
\centering
\includegraphics[width=0.8\textwidth]{figures/prog_languages.pdf}
\caption{Programming languages used for the Top 5 social media platforms}
\label{fig:languages}
\end{figure}
\paragraph{Geospatial analysis:}
\texttt{Github} and \texttt{Gitlab} allow each user to specify their respective geolocation. As it was already indicated in \cite{Kollanyi2016}, the platforms use a free text field for collecting the information from the user. Therefore, we can not assume standardized data for our analysis. We utilized Google's geocoding API to receive longitude/latitude pairs that are approximately close to the location that was specified by the user. In total, we gathered information of 46.900 unique contributors, where geolocation information was present in 22.688 cases.
\begin{figure}[tb]
\centering
\includegraphics[width=\textwidth]{figures/twitter_map.pdf}
\caption{Origin of Twitter repositories }
\label{fig:twitter}
\end{figure}
Kollanyi \cite{Kollanyi2016} already investigated the global distribution of twitter bot repositories. We present an updated version of the map with a similar distribution that was already observed in 2016. As one can see on the world map in Figure \ref{fig:twitter} the main part of the repositories belong to ten countries. This observation is comparable to the findings of \cite{Kollanyi2016}. Having a look at the top five most contributing countries and comparing them to the findings of \cite{Kollanyi2016} from 2016, shows, that the distribution basically stays the same.
Most of the Twitter repositories originate from the United States. In \cite{Kollanyi2016}
the United States were directly followed by Japan. Our updated version reveals that United Kingdom caught up to Japan and follows the U.S. by providing the second largest number of bot related repositories for Twitter. In contrast to Kollanyis study we also have access to location data of different social-media platforms. Directly compared to the distribution of Twitter, we observe some inherent dissimilarities between the platforms. While Russia does not play an important role in the context of Twitter bots, most of the Telegram bot code contributors are from that country (Figure \ref{fig:telegram}). A reason for this could be the popularity of Telegram within the Russian population~\cite{nytimestelegram}.
Whatsapp contributors mainly originate from India, where the messenger is not only used for private communication, but serves as a central communication and information channel~\cite{StatistaIndia2016}.
\begin{figure}[h!]
\centering
\includegraphics[width=\textwidth]{figures/telegram_map.pdf}
\caption{Origin of Telegram repositories }
\label{fig:telegram}
\end{figure}
\section{Content Analysis}
\label{sec:qual_analysis}
Identifying the capabilities, operational scenarios, and associated costs of automated bot programs in the context of social media platforms is regarded as one of the major goals of this study. Therefore -- on a qualitative level -- the content as well as the overall topics of available social bot code is of central interest.
Since, in general, manual inspection of the code base or the description of each repository is unfeasible for the number of gathered repositories, we utilize unsupervised learning techniques to identify topic trends over time in an exploratory way. We start with an additional and explicit content analysis of gathered repositories. We apply dominance filtering from decision theory in order to bound the number of 'interesting' repositories for manually detailed investigation.
\paragraph{Analysis of (interesting) Content:}
\label{sec:multiobjective}
To identify a subset of interesting repositories $R'\subseteq R$ for manual inspection, we specify three different indicators of interest: $N_c(r)$, the \emph{number of commits}, $L(r)$ the \emph{repository lifespan}, and $T(r)$ \emph{repository timeliness} with $r\in R$. While the number of commits is a rough indicator for overall user engagement in a software project (represented by the repository), the lifespan as defined in section~\ref{sec:descr_analysis} provides insights into the age of the respective repository. Further, we measure the timeliness of a repository as the difference between the current time and the time of the last update. Hence, a low timeliness value indicates a repository that has recently been updated, whereas a high value represents a repository that has not been updated for a long time and is probably inactive.
Using the defined simple indicators, any repository $r\in R$ can be represented by a three dimensional vector $v(r) = (N_c(r), L(r), T(r))^T\in \mathbb{N}^3$.
In our interpretation of these three indicators, an ideal (i.e. most interesting) repository for manual analysis would have large $N_c$, large $L$, and small $A$ components. Clearly, multiple dimensions of interest can result in incomparable repositories, such that no strict order of interest can be achieved. To identify the most promising trade-offs regarding the three indicators, we employ the notion of dominance, usually applied in the field of decision theory and multiobjective optimization. We say a $d$-dimensional vector $v$ dominates another $d$-dimensional vector $w$ (notation $v \prec w$), if and only if the following conditions hold: (1) $\forall i \in {1,\dots,d}$ : $v_i \leq w_i$ and (2) $\exists i \in {1,\dots,d}$ : $v_i < w_i$.
Note, that $\le$ and $<$ are interpreted as \emph{scalar domination}, i.e., that this relation is interpreted differently for maximization and minimization.\footnote{To overcome the distinction of minimization and maximization, we can convert minimizing indicators to maximizing indicators (or vice versa) due to the duality principle.}
In other words, a repository only dominates another repository, if for each indicator of interest, the first repository performs at least equally well as the other one and for at least one indicator strictly better. If this is not the case, we call both repositories incomparable or non-dominating to each other.
To eventually select the dominant (and thus most interesting) repositories for manual content analysis, we perform a pairwise comparison of the indicator vectors of all repositories from $R$ and remove those, which are dominated (non-domination filtering). As a result, we acquire $83$ incomparable and non-dominated repositories forming $R'$ for further analysis.
The analysis is done manually, by reading through repository descriptions and read me files. The results show, that most of the repositories yield code for the social media platforms Telegram (33), Twitter (27), and Reddit (11), reflecting the overall distribution of all bot codes (see Figure \ref{fig:searchterm_dist}). There exist three repositories containing code for the platform Instagram and one repository per each of Whatsapp, Facebook, Youtube, and Skype\footnote{Five of the repositories were omitted, since they contain e.g. Chinese descriptions or are related to other topics.}.
Table \ref{tab:support-correlation} shows the functionality and purpose, which is handled by the repositories. Most of the repositories can be grouped, since they provide similar functionality.
\begin{table}[!h]\small
\begin{center}
\begin{tabular}{@{}llr@{}}
\toprule
\textbf{Type}& \texttt{Functions}&\texttt{Number of Repos} \\ \midrule
Native Social Bot Functionality & posting, liking, &22\\ & following & \\
Wrapper/ Framework/ Library & easy API access, &27\\ & pre-defined functions & \\
Other Service Functionality & downloading data,& 14\\& payment handling, & \\ & include external services & \\
\bottomrule
\end{tabular}
\end{center}
\caption{Functionality of Top Repositories}
\label{tab:support-correlation}
\end{table}
Twenty-two of the top repositories contain code which enables the user to implement social bot activities. Exemplary actions are posting predefined content, building follower networks, or liking specific hashtags. Most of the repositories contain code, which enables users to run their own bot script by adjusting a few lines of code. Related to the bot taxonomy of Grimme et. al \cite{Grimme2017}, we especially find bot repositories, which can be assigned to the class of simple bots. The 22 repositories, which are part of the two non-dominated layers, contain simple API supported functions like posting, liking or following. None of the code repositories provided code for "intelligently" acting bots.
Apart from the native social bot functionality, the main share of repositories contains code in order to handle API access for a specific programming language or to provide predefined code libraries. These 27 repositories can be applied by users, which are willing to create their own bot programs. In contrast to the repository type described above, these repositories do not provide off-the-shelf executable code for a specific bot program. Rather, they simplify the usage of programming interfaces for different programming languages.
Next to frameworks and API wrappers, there exists a number of dedicated bot scripts. The scripts contain code, which enables the user to establish own bots on a specific platforms. The analyzed repositories provided bots, which are able to fulfill simple tasks like following, liking or sharing posts. Most often a detailed setup description is provided and the code is easy to customize for own purposes.
About 14 repositories provide code, which fulfills no common social bot functions. These repositories contain scripts, which expand the platform by additional functionality. Examples are functions for downloading data, adding payment options or building interfaces for the inclusion of external services like Open Street Map.
\paragraph{Topic Modeling:}
Topic-modeling is used in the context of natural language processing to automatically extract semantic meaning from a given data corpus by identifying similarities between documents and grouping them together. We consider the application of one of the most frequently used algorithm in the field of natural language processing: Latent Dirichlet Allocation (LDA)\cite{lda}. In LDA, a generative statistical model is used to describe documents as a set of latent topics, where each topic follows a unique word distribution, which in turn is generated by maximizing the conditional probability for a word to occur in a given topic. While LDA usually achieves excellent results for static corpora, i.e. data that does not change over time, it offers very limited insights when it comes to topic evolution over time. However, for the given analysis it is crucial to understand trends in the context of social bot creation and how and when these trends evolve.
Given the requirements mentioned before, we decide to additionally employ a different method for topic modelling which is able to track changes over time: Stream Clustering. While clustering in general is an unsupervised machine learning technique that aims to identify homogeneous groups of observations, stream clustering expands upon this idea and solves different shortcomings of the traditional approaches. First, traditional clustering algorithms have to iterate over the data multiple times. This is not feasible for large and potentially unbounded data sources. Furthermore, stream clustering algorithms provide mechanisms to decay clusters over time and therefore account for changes within the underlying data distribution (concept drift). In the context of this work, we utilize stream clustering mainly for dealing with concept drift and identifying trends within the bot creation community. Specifically, we interpret the crawled repositories as a discontinuous data stream over time and use the method \texttt{textClust} to cluster the repositories descriptions.
Since the main goal of our analysis is the identification of time-depended trends and topics based on the repository description, a few pre-processing steps need to be conducted. In a first step, we select all repositories that have English descriptions with more than one word. Additionally, we remove all occurrences of social-media platform names from the documents, since the goal of the analysis is the identification of bot code functionality rather than platforms. Next, we apply common data transformation steps such as tokenization, stop word removal and lemmatization. In addition to unigrams, we also create bi-grams to account for close relationships between words within one document, such as word combinations like \textit{markov chain}. Based on the pre-processed data, we execute a LDA analysis~\cite{Rehurek2010}. Manual inspection of the results shows that about 15 topics are well suited to reveal existing repository types. Allowing a larger number of clusters leads to artificial separation of topics, while a smaller number of allowed clusters leads to overarching topics that contain multiple, semantically rather different repository types.
\begin{table}\small
\centering
\caption{Top eleven (out of fifteen) topic representatives provided by LDA}
\label{tab:lda}
\begin{tabular}{@{}ll@{}}
\toprule
Topic & Words \\ \midrule
0 & \textbf{chat}, message, group, send, user, app, via, google, friend, bot \\
1 & \textbf{API}, using, written, python, \textbf{framework}, library, php, create, use, written\_python \\
2 & python, script, first, learning, small, price, twitterbot, bot, reddit, tutorial \\
3 & tweet, random, reply, test, \textbf{markov}, text, chain, generates, markov\_chain, given \\
4 & platform, slack, implementation, ruby, language, answer, question, messenger \\
5 & manage, aws, telegrambot, play, game, url, notification, lambda, world, live \\
6 & \textbf{simple}, bot, weather, people, creating, thing, sample, heroku, program, template \\
7 & tweet, word, every, sends, day, info, picture, give, hour, random \\
8 & news, game, user, service, follow, follower, help, automated, card, profile \\
9 & post, nodejs, tweet, account, twitter, search, using, image, user, made \\
10 & \textbf{chatbot}, example, quote, personal, based, schedule, assistant, daily, bot, working \\\bottomrule
\end{tabular}
\end{table}
Table \ref{tab:lda} lists the resulting first eleven topics, which confirm our findings from the decision making and analysis process in Section \ref{sec:multiobjective}. Most of the topics represent repositories that provide simple functionality or user action such as posting random content (e.g. images or predefined messages), linking videos, or following other users (5, 6, 7, 8, 9). Besides, some clusters describe more sophisticated functionality, which enables interaction between different accounts like chatting with other users (0, 3, 10). In this context, we observe that Markov chains play an important role. However, the top representatives of the resulting topics do not indicate that state of the art machine learning algorithms are utilized. Cluster one represents repositories that provide frameworks or use existing platform APIs.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{figures/topics.pdf}
\caption{Most important micro-cluster over time.}
\label{fig:micro-cluster}
\end{figure}
In a last step, we apply the stream clustering approach to inspect how the observed repository types behave over time, i.e. reveal potential trends within the bot-code community. As already mentioned before, we utilize a dedicated text-based stream clustering algorithm called \texttt{textClust}\cite{ER17Twitch}.
In general, the algorithm follows a two step approach, which is commonly used in the context of stream clustering. First, an online component summarizes the incoming text stream (represented as numerical \texttt{TF-IDF} vectors) in an online fashion, which results in a set of micro-clusters. These micro-clusters are considered as dense areas in data space. Periodically, micro-clusters are revisited and older entries which are not updated recently are decayed and ultimately removed. While micro-clusters are created, updated, and removed during the online-phase, they are aggregated macro level during the re-clustering phase. Therefore, traditional clustering algorithms can be used to aggregate the different micro-clusters. For our analysis, however, we focus on micro-clusters since they represent topics of repositories in our case. The algorithm is executed with default parameter settings. A pseudo-code description of the update and cleanup procedures of \texttt{TextClust} is provided in the Apendix, see Algorithms~\ref{Algorithm:update} and~\ref{Algorithm:cleanup}. Since the data-input is not uniformly distributed over time, we also select the algorithm's time variant, which fades data based on a predefined timedelta, instead of the number of observations. An in-depth description of the algorithm can be inspected in \cite{ER17Twitch}\footnote{The code can be accessed at \url{https://wiwi-gitlab.uni-muenster.de/stream/textClust}}.
Within Figure \ref{fig:micro-cluster} the weight of micro-clusters containing specific terms are displayed. In addition to important topic terms from the LDA analysis, we also inspected micro-clusters containing machine learning related tokens. The results of our stream clustering approach give some indication of topic importance (in terms of cluster weight over time). Micro-clusters containing the \textit{simple} term have the highest cumulative weight and therefore are dominant topics during the observed time-span. As it was already discussed, repositories that are assigned to those micro-clusters mainly implement simple functionalities such as automatic following, posting or displaying the weather forecast using external data. Clearly, machine learning repositories only play a minor role with a low overall weight, compared to the other cluster terms and represent a rather new trend that arose in early 2017. While most micro-clusters increase their weight over time in a rather linear fashion, the API term makes an exception with a significant weight increase in July 2015. As it was already analyzed before, this is due to the fact that Telegram published their bot API, one month before. Hence, we see that external events can directly impact the bot community. Since machine/deep learning was not reflected as a distinct topic within our LDA analysis (due to the small number of repositories containing these terms), we use our stream clustering results to get an understanding of important terms, related to that topic. Therefore we look at micro-clusters containing either the term machine learning or deep learning and all tokens that are assigned to the similar cluster. Figure \ref{fig:deep1} shows the results as a word cloud. It indicates that machine learning techniques are mainly used in context of chat bots, i.e. the creation of human-like messages. However, also detection mechanisms such as hate-speech identification are present.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{figures/deep1.pdf}
\caption{Most frequent tokens in micro-clusters containing the term \texttt{deep learning}}
\label{fig:deep1}
\end{figure}
Furthermore, we manually extracted and analyzed repositories, which contain keywords like machine learning, deep learning or artificial intelligence within their description. As a result we identified and inspected 85 distinct repositories manually. However, having a look at the descriptions and read-me files, it turns out, that most of the machine-learning-related repositories (17) contain code for social bot detection. The contributors report on using classification approaches like Support Vector Machines or Logistic Regression in order to distinguish between human and bot steered accounts. Next to bot detection, content detection, e.g., in terms of detecting hateful or harmful content, is covered by the analyzed repositories (14).
A large number of repositories (15) is related to chat functionality. Contributors use machine learning, deep learning and AI techniques to create chat bots, which are able to automatically answer to messages. Other repositories (14) create new posts on basis of former posts or an external text data base. Here natural language processing techniques are used to imitate specific writing styles.
Only two of the 85 repositories are described as "stand alone" bot scripts, which could be implemented to replace a human user. It is important to mention, that those scripts do not provide code for intelligent bots like introduced in e.g. the taxonomy of Grimme et al. Rather, they include some language processing techniques, which e.g. imitate a specific writing style for posts, or filter by the use of e.g. classification for interesting content to follow/like/share.
Next to the described functionality, there exist a number of repositories (8), which simply name keywords like "Machine Learning" or "AI" within their description, but do not contain machine learning, deep learning or AI technologies at all.
The remaining 15 repositories are either not further described, are already outdated, or contain descriptions of other languages.
\section{Discussion}
\label{sec:discussion}
With this work, we provide a comprehensive study on the availability and state of development of open program code, which can be utilized for realizing the automation of accounts in social media. Based on developer repositories in the most relevant open collaboration platforms (mainly Github, Gitlab, and Bitbucket), we analyzed software development for a broad spectrum of social media platforms using various knowledge discovery techniques that range from descriptive analysis to stream clustering for trend detection. While the descriptive analysis proves overall availability of projects dealing with the development of social bots, a partition of data with respect to social media platforms provided additional insights into the importance and accessibility (and thus possible resilience) of these platforms for (against) simple automation of accounts. The performed geospatial analysis offers insights into regional importance of platforms and may point to main targets for social bot application at these locations. This results in different levels of potential "threat" in the world for different platforms as a consequence of user preferences. A specific view on programming languages, however, indirectly tells us some details on the operators and programmers of social bots. The overwhelmingly dominant usage of interpreted programming languages (Python and Java Script) allow simple and rapid development of code also for inexperienced developers. This suggests that a large proportion of the observed projects is developed to enable or simplify usage of social bots or the realization of data acquisition from social media platforms at a prototype level. Additionally, the availability of simple API frameworks for specific social media platforms implies the use of simple programming languages like Python, while Java Script is presumably used to access data and functionality which is not reachable via the APIs.
The qualitative analysis of social bot projects is based on relevance of repositories, which was determined by a decision theoretical approach considering multi-criteria domination regarding repository life span, timeliness, and number of commits. The analysis of these repositories showed that the predominant functionalities provided in those repositories are restricted to simple bot development tools, frameworks, or specific services not specifically related to social bots alone.
While these findings suggest, that openly available social bot technology is of rather simple nature, the topic modelling via LDA and stream clustering provides deeper insights into relevant types of social bot-related development as well as into the change of development focus over time. These aspects specifically address research question two of this work: We do not find ready-to-use software that exceeds simple functionality like (re-)posting prepared content, favoring articles/posts, or following other accounts in an automated manner. Also, the most important topics extracted by stream clustering suggest, that further functionality is restricted to building blocks for social bot construction, chat bot facilities, and API wrappers or alternatives / bypasses. Interestingly, machine learning (related to artificial intelligence) is an existing but neglectable topic in open social bot development. It mainly appears in the context of social bot detection. Sophisticated or ready-to-use mechanisms for the implementation of 'intelligent bots' are not available.
The observations of the descriptive and qualitative analysis together imply that we can distinguish two scenarios for the costs of social bot development (third research question). The simple scenario is the one where operators only aim for trivial usage of the available APIs of social media platforms performing actions like posting, favouring, or sharing. For these purposes, off-the-shelf software is freely available and feasible. Additionally, there is a set of proprietary interfaces and frameworks to easily enable such tasks. Considering very simple cases of content distribution, the realization of this type of social bots is relatively cheap. More expensive but probably in most cases still feasible is the realization of advanced social bots that simulate human behavior on the activity level - not in interaction with others. Available and open (some even well established and continuously maintained) social bot frameworks enable the enrichment of ready-to-use building blocks with more complex code. Experiments of Grimme et al.~\cite{Grimme2017,Grimme2018} demonstrated, that such extensions are feasible and require only medium technical expertise. A major gap, however, can be identified between the sometimes postulated existence of intelligently acting bots -- i.e. bots which are able to produce original content (related to a defined topic), provide reasonable answer to comments, or even discuss topics with other users -- and available open software components. The absence of tools implies that either the development of such techniques is too difficult and too costly to be provided open for the public, or that these techniques are scientifically too advanced to be subject of current social bot development. Current reports on experimental intelligent social bots by large software companies, however, suggest that there is currently no productive 'intelligent' software available~\cite{wptay2016,wiredzo2017}. Others observations show that 'intelligence' on conversational level is (intentionally) restricted to advanced chat bot capabilities~\cite{quartzzo2018}, as simple learning approaches are too sensitive regarding external manipulation. As such, the costs for realizing 'intelligent' bots can be considered infeasible, today.
Overall, this work and its analysis provides a new foundation for further discussion on the existence and influence of social bots extracted from an extensive evaluation of data on current open software projects. Certainly, software development is an ongoing process and technologies will change or advance together with advances in science (e.g. new developments in artificial intelligence) or with modifications in the technological environment (e.g. changes in APIs or accessibility of social media platforms). This paper provides a methodological framework for comprehensive analysis that goes beyond a singular investigation. It can be applied over time to monitor the development of technology and application of social bots.
\paragraph{Limitations and Future Work:}
Despite from the fact that our work reveals comprehensive insights into the bot programming community, some limitations still exist. One of the main limitations of this work is the restriction on open-bot code. It could be argued that social bots are frequently used in context of user manipulation. Therefore sophisticated bot programs which work well on that manipulation task, exhibit potential monetary value and thus could be sold to interested third parties. Hence, a platform which is based upon the idea of open-source code distribution may not be the right place to host the program code. In order to overcome this issue, a supplementary study, focusing on dedicated marketplaces in the dark web, could be conducted. Another limitation of our study is missing functionality verification. Since the source code of the different repositories was not explicitly executed, we cannot verify whether the code actually implements the functionality that was described in the repository's metadata. While for simple functionality such as automatic procedures to follow persons or liking their posts, we expect that the description actually reflects the described services, we cannot expect this from repositories that do apply complex machine learning algorithms and offer full-service bot programs. Lastly, we may not captured all social bot code that is currently available on the open-source platforms because they were not found by our predefined query.
As we showed within our analysis, the community of bot code is constantly changing over time. Bot code programmers highly depend on the social platforms and their support for external programming interfaces while the platforms themselves constantly adjusting their policy and therefore also their API's because of public pressure. Furthermore new technologies such as machine- or deep learning emerge in the field and are more frequently utilized. Although we currently do not observe any sign that intelligent bots, as they were described in Section \ref{sec:related_work}, we do see a need for monitoring the development regularly.
\newpage
\bibliographystyle{ieeetr}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,498 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/3038\/how-to-test-hypothesis-of-no-group-differences?noredirect=1","text":"# How to test hypothesis of no group differences?\n\nImagine you have a study with two groups (e.g., males and females) looking at a numeric dependent variable (e.g., intelligence test scores) and you have the hypothesis that there are no group differences.\n\nQuestion:\n\n\u2022 What is a good way to test whether there are no group differences?\n\u2022 How would you determine the sample size needed to adequately test for no group differences?\n\nInitial Thoughts:\n\n\u2022 It would not be enough to do a standard t-test because a failure to reject the null hypothesis does not mean that the parameter of interest is equal or close to zero; this is particularly the case with small samples.\n\u2022 I could look at the 95% confidence interval and check that all values are within a sufficiently small range; perhaps plus or minus 0.3 standard deviations.\n\u2022 what do you mean by \"this assumes the null hypothesis to be true\" ? \u2013\u00a0robin girard Sep 24 '10 at 8:39\n\u2022 If you want to be able to control the probability of declaring wrongly \"there is a difference\" you need to separate the two hypothesis (did I already mentionned I love this quote: stats.stackexchange.com\/questions\/726\/\u2026 ;) ) \u2013\u00a0robin girard Sep 24 '10 at 8:40\n\u2022 @Robin the p value of a null hypothesis significance test is the probability of seeing as or more extreme data than that observed assuming the null hypothesis is true; but perhaps I could word the statement above better. \u2013\u00a0Jeromy Anglim Sep 24 '10 at 10:26\n\u2022 @Robin I modified the question to try to make my point clearer \u2013\u00a0Jeromy Anglim Sep 24 '10 at 10:32\n\nI think you are asking about testing for equivalence. Essentially you need to decide how large a difference is acceptable for you to still conclude that the two groups are effectively equivalent. That decision defines the 95% (or other) confidence interval limits, and sample size calculations are made on this basis.\n\nThere is a whole book on the topic.\n\nA very common clinical \"equivalent\" of equivalence tests is a non-inferiority test\/ trial. In this case you \"prefer\" one group over the other (an established treatment) and design your test to show that the new treatment is not inferior to the established treatment at some level of statistical evidence.\n\nI think I need to credit Harvey Motulsky for the GraphPad.com site (under \"Library\").\n\nBesides the already mentioned possibility of some kind of equivalence test, of which most of them, to the best of my knowledge, are mostly routed in the good old frequentist tradition, there is the possibility of conducting tests which really provide a quantification of evidence in favor of a null-hyptheses, namely bayesian tests.\n\nAn implementation of a bayesian t-test can be found here: Wetzels, R., Raaijmakers, J. G. W., Jakab, E., & Wagenmakers, E.-J. (2009). How to quantify support for and against the null hypothesis: A flexible WinBUGS implementation of a default Bayesian t-test. Psychonomic Bulletin & Review, 16, 752-760.\n\nThere is also a tutorial on how to do all this in R:\n\nhttp:\/\/www.ruudwetzels.com\/index.php?src=SDtest\n\nAn alternative (perhaps more modern approach) of a Bayesian t-test is provided (with code) in this paper by Kruschke:\n\nKruschke, J. K. (2013). Bayesian estimation supersedes the t test. Journal of Experimental Psychology: General, 142(2), 573\u2013603. doi:10.1037\/a0029146\n\nAll props for this answer (before the addition of Kruschke) should go to my colleague David Kellen. I stole his answer from this question.\n\n\u2022 I was wondering if someone would provide a Bayesian approach. Excellent. Thanks. \u2013\u00a0Jeromy Anglim Sep 24 '10 at 11:41\n\u2022 It may be worth updating this answer to include a reference to the awesome BayesFactor package for R. \u2013\u00a0crsh Apr 27 '17 at 16:22\n\nFollowing Thylacoleo's answer, I did a little research.\n\nThe equivalence package in R has the tost() function.\n\nSee Robinson and Frose (2004) \"Model validation using equivalence tests\" for more info.\n\n\u2022 Thanks for the link and the pointer to the equivalence package. \u2013\u00a0chl Sep 26 '10 at 9:59\n\nThere are a few papers I know of that could be helpful to you:\n\nTryon, W. W. (2001). Evaluating statistical difference, equivalence, and indeterminacy using inferential confidence intervals: An integrated alternative method of conducting null hypothesis statistical tests. Psychological Methods, 6, 371-386. (FREE PDF)\n\nAnd a correction:\nTryon, W. W., & Lewis, C. (2008). An Inferential Confidence Interval Method of Establishing Statistical Equivalence That Corrects Tryon\u2019s (2001) Reduction Factor. Psychological Methods, 13, 272-278. (FREE PDF)\n\nFurthermore:\n\nSeaman, M. A. & Serlin, R. C. (1998). Equivalence confidence intervals for two-group comparisons of means. Psychological Methods, Vol 3(4), 403-411.\n\n\u2022 There are tons of papers and even books on this topic. \u2013\u00a0Michael Chernick May 6 '12 at 4:53\n\nI have recently thought about an alternative way of \"equivalence testing\" based on a distance between the two distributions rather than between their means.\n\nThere are some methods providing confidence intervals for the overlap of two Gaussian distributions:\n\nThe overlap $O(P_1,P_2)$ of (between?) two distributions $P_1$ and $P_2$ has a nice probabilistic interpretation: $$1-O(P_1,P_2)= TV(P_1,P_2)$$ where $TV(P_1,P_2) = \\sup_A \\big|P_1(A) - P_2(A) \\big|$ is the total variation distance between $P_1$ and $P_2$.\n\nThat means that, for example, if $O(P_1,P_2)>0.9$ then the probabilities given by $P_1$ and $P_2$ of any event do not differ more than $0.1$. Roughly speaking, the two distributions make the same predictions up to $10\\%$.\n\nThus, instead of using an acceptance criterion based on a critical value for the difference between the means $\\mu_1$ and $\\mu_2$, as in classical equivalence testing, we could base it on a critical value for the difference between the probabilities of the predictions given by the two distributions.\n\nI think there's an advantage in terms of \"objectiveness\" of the criterion. The critical value of $|\\mu_1 - \\mu_2|$ should be given by an expert of the real problem: this should be a value beyond which the difference has a practical importance. But sometimes nobody has a solid knowledge about the real problem and there's no expert able to provide a critical value. Adopting a conventional critical value about $TV(P_1,P_2)$ could be a way to a criterion not depending on the physical problem under consideration.\n\nIn the Gaussian case with same variances, the overlap is one-to-one related to the standardized mean difference $\\frac{|\\mu_1-\\mu_2|}{\\sigma}$.\n\n\u2022 Do you have any resources showing overlap being used in some real problems? This sounds incredibly promising, but it's not clear to me how one would apply it in a real problem (where your conclusions are potentially several steps removed from \"this distribution is pretty similar to X\", thereby making it a little hard to see how that 10% TV translates into size of impact on inferences). \u2013\u00a0Stumpy Joe Pete Jul 8 '15 at 7:16\n\u2022 @StumpyJoePete I have written something in the same spirit on my blog: stla.github.io\/stlapblog\/posts\/\u2026 \u2013\u00a0St\u00e9phane Laurent Jul 8 '15 at 8:15\n\nIn the medical sciences, it is preferable to use a confidence interval approach as opposed to two one-sided tests (tost). I also recommend graphing the point estimates, CIs, and a priori-determined equivalence margins to make things very clear.\n\nYour question would likely be addressed by such an approach.\n\nThe CONSORT guidelines for non-inferiority\/equivalence studies are quite useful in this regard.\n\nSee Piaggio G, Elbourne DR, Altman DG, Pocock SJ, Evans SJ, and CONSORT Group. Reporting of noninferiority and equivalence randomized trials: an extension of the CONSORT statement. JAMA. 2006, Mar 8;295(10):1152-60. (Link to full text.)\n\n\u2022 I wouldn't necessarily say that confidence intervals are preferred. In fact the confidence intervals correspond to hypothesis tests. TOST can be achieved by looking at the confidence intervals obtained by intersecting the two one sided confidence intervals that correspond to the two one-sided t tests that are used in the procedure. \u2013\u00a0Michael Chernick May 6 '12 at 4:59\n\nYes. This is equivalence testing. Basically you reverse the null and alternative hypothesis and base the sample size on the power to show that the difference of the means is within the window of equivalence. Blackwelder called it \"Proving the null hypothesis.\" This is commonly done in pharmaceutical clinical trials where equivalence of a generic drug to the marketed drug is tested or an approved drug is compared to a new formulation (often called bioequivalence). The one sided version is called non-inferiority. Some times a drug can be approved by just showing that the new drug is not inferior to the marketed competitor. Shao and Pigeot have developed a consistent bootstrap approach to bioequivalence using crossover designs.\n\nBootstrap differences (e.g. the difference between the means) between the 2 sample groups and check for statistical significance. A more detailed description of this approach, albeit in a different context, can be found here http:\/\/www.automated-trading-system.com\/a-different-application-of-the-bootstrap\/\n\n\u2022 You are conflating the fallacy of accepting the null hypothesis of no difference and finding evidence that two quantities are equivalent. \u2013\u00a0Alexis May 4 '14 at 15:00","date":"2019-02-20 01:29:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.5540274977684021, \"perplexity\": 888.7253732050561}, \"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-2019-09\/segments\/1550247494125.62\/warc\/CC-MAIN-20190220003821-20190220025821-00078.warc.gz\"}"} | null | null |
{"url":"http:\/\/openqemist.1qbit.com\/docs\/rigetti_example.html","text":"# OpenQEMIST and Rigetti example\u00b6\n\nThis notebook shows how OpenQEMIST can be combined with the Rigetti stack to use the Variational Quantum Eigensolver (VQE) as an electronic structure solver, and combine it with a problem-decomposition technique such as Density Matrix Embedding Theory (DMET).\n\n## VQE Example\u00b6\n\nThis tutorial assumes that the user has correctly set up and configured the OpenQEMIST package. The Variational Quantum Eigensolver (VQE):math:^{1,2} is a hybrid quantum-classical algorithm for simulating quantum systems. We here focus on VQE within the context of solving the molecular electronic structure problem for the ground-state energy of a molecular system. In VQE, we first prepare the trial wavefunction (quantum state) $$\\vert \\Psi(\\vec{\\theta}) \\rangle = U(\\vec{\\theta}) \\vert 0 \\rangle$$ based on an ansatz that depends on $$m$$ parameters defining $$\\vec{\\theta}=(\\theta_1, \\theta_2, \\ldots, \\theta_m)$$. The expectation value of the Hamiltonian ($$\\hat{H}$$), $$\\langle \\Psi(\\vec{\\theta}) \\vert \\hat{H} \\vert \\Psi(\\vec{\\theta}) \\rangle$$, will then be simulated.\n\nThe expectation value can be minimized based on the variational principle,\n\nbegin{equation} E = min_{vec{theta}} frac{langle Psi(vec{theta}) vert hat{H} vert Psi(vec{theta}) rangle}{langle Psi(vec{theta}) vert Psi(vec{theta}) rangle} geq E_{text{gs}}nonumber end{equation}\n\nwhich ensures that the energy computed will be an upper bound to the true ground-state energy $$E_{\\text{gs}}$$. This allows us using classical minimizers to find optimal parameters $$\\vec{\\theta}$$ for the ground-state energy $$E_{\\text{gs}}$$.\n\nVQE can be performed using OpenQEMIST in conjuction with the Rigetti stack for calculating the ground state energy of a molecular system. The unitary coupled-cluster ansatz can be used to prepare the trial wavefunction $$\\vert \\Psi(\\vec{\\theta}) \\rangle$$. In this notebook, we will show you an example using a small molecule, the hydrogen molecule (H:math:_text{2}), for a simulation using VQE.\n\nfrom openqemist.electronic_structure_solvers import VQESolver, FCISolver\nfrom openqemist.quantum_solvers import RigettiParametricSolver\n\nfrom pyscf import gto\n\n# Build the molecule\nH2 = [['H', [0.0, 0.0, 0.0]], ['H', [0.0, 0.0, 0.74137727]]]\nmol = gto.Mole()\nmol.atom = H2\nmol.basis = \"sto-3g\"\nmol.charge = 0\nmol.spin = 0\nmol.build()\n\n# Configure the solver object\nvqe_solver = VQESolver()\nvqe_solver.hardware_backend_type = RigettiParametricSolver\nvqe_solver.ansatz_type = RigettiParametricSolver.Ansatze.UCCSD\n\n\nWe can now simulate the molecule and get its energy.\n\nenergy_fci = FCISolver().simulate(mol)\nenergy_vqe = vqe_solver.simulate(mol)\n\nprint(\"\\nFCI energy = \", energy_fci)\nprint(\"VQE energy = \", energy_vqe)\n\nFCI energy = -1.1372704220924401\nVQE energy = -1.1372704138513532\n\n\nIt is possible to use different initial parameters for the optimization:\n\n# Using custom initial parameters\n# Getting the dimension of the initial parameters vector\nnum_var_params = vqe_solver.hardware_backend.amplitude_dimension\n# Set the intial parameters for the solver\nvqe_solver.initial_var_params = [0.01 for i in range(num_var_params)]\n\nvqe_solver.simulate(mol)\n\nVQE : initial variational parameters:\n[0.01, 0.01]\n\n-1.1372665775495083\n\n\n## Using the QVM shot-based simulator\u00b6\n\nTo use the QVM, we can use the backend_parameters attribute of the VQESolver object. The VQE object then configures the hardware backend automatically. Because the QVM is slower than the default wavefunction simulator backend, we specify an optimizer function that returns after a few iterations, in the interest of showing the usage of the solver in a reasonable time. See the documentation for more details about using custom optimizers. This interface is what would also be used to target a QPU backend in the future.\n\ndef quick_optimizer(backend, amplitudes):\nfrom scipy.optimize import minimize\n\nprint(\"Running using custom optimizer.\")\n\n# We use a force the optimizer to return after 2 iterations.\nresult = minimize(backend, amplitudes, method='COBYLA',\noptions={'disp':True, 'maxiter':2})\n\nreturn result.fun\n\nvqe = VQESolver()\nvqe.optimizer = quick_optimizer\n\n\nTo use the QVM, we can use the backend_parameters attribute of the VQESolver object. The VQE object then configures the hardware backend automatically. We can then run the simulation with the object. The number of shots can also be set with this parameter.\n\nNote that because we restricted the optimizer to 2 iterations and reduced the number of shots, the resulting energy will not be accurate.\n\nvqe.hardware_backend_type = RigettiParametricSolver\nvqe.ansatz_type = RigettiParametricSolver.Ansatze.UCCSD\nvqe.backend_parameters = {'backend': '4q-qvm', 'n_shots': 10}\n\nenergy = vqe.simulate(mol)\nprint(\"Unconverged QMV energy: \", energy)\n\nUnconverged QMV energy: -1.146711378495224\n\n\n## DMET Example\u00b6\n\nAt the current early stage of quantum hardware, the available computational resource is yet very limited. Thus, it is still challenging to perform accurate electronic structure calculations on actual quantum hardware. Simulation on classical computer requires large computational cost as well. Therefore, we need to reduce the problem size while maintaining the accuracy of electronic structure calculation to solve a problem for small sized molecules to perform quantum simulations.\n\nDensity Matrix Embedding Theory (DMET):math:^{3,4} is a powerful problem decomposition technique to reduce the problem size, while maintaining the accuracy of the electronic structure calculation. The DMET method decomposes a molecule into fragments, and each fragment is treated as an open quantum system that is entangled with each of the other fragments, all taken together to be that fragment\u2019s surrounding environment (or \u201cbath\u201d). VQE algorithm can be used with DMET using OpenQEMIST in conjuction with the Rigetti stack.\n\nIn this notebook, we will show you an example of H$$_\\text{4}$$ molecule for DMET simulation using VQE as an electronic structure solver.\n\nfrom openqemist.problem_decomposition import DMETProblemDecomposition\nfrom openqemist.problem_decomposition.electron_localization import meta_lowdin_localization\n\nH4 = [['H', [0.7071067811865476, 0.0, 0.0]],\n['H', [0.0, 0.7071067811865476, 0.0]],\n['H', [-1.0071067811865476, 0.0, 0.0]],\n['H', [0.0, -1.0071067811865476, 0.0]]]\n\nmol = gto.Mole()\nmol.atom = H4\nmol.basis = \"minao\"\nmol.charge = 0\nmol.spin = 0\nmol.build()\n\ndmet = DMETProblemDecomposition()\ndmet.verbose = True\ndmet.electron_localization_method = meta_lowdin_localization\n# Set the DMET object to use the solver that we configured above\ndmet.electronic_structure_solver = vqe_solver\nenergy_vqe = dmet.simulate(mol, [1,1,1,1])\n\nprint(\"The DMET energy is: \", energy_vqe)\n\nThe DMET energy is: -1.9916111022655567\n\n\n## References\u00b6\n\n1. Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Al\u00e1n Aspuru-Guzik, and Jeremy L. O\u2019Brien, \u201cA variational eigenvalue solver on a photonic quantum processor\u201d, Nat. Commun., 5, 4213 (2014).\n\n2. Jarrod R. McClean, Jonathan Romero, Ryan Babbush, and Al\u00e1n Aspuru-Guzik, \u201cThe theory of variational hybrid quantum-classical algorithms\u201d, New J. Phys., 18, 023023 (2016).\n\n3. Gerald Knizia and Garnet K.-L. Chan, \u201cDensity Matrix Embedding: A Simple Alternative to Dynamical Mean-Field Theory\u201d, Phys. Rev.\u00a0Lett., 109, 186404 (2012).\n\n4. Sebastian Wouters, Carlos A. Jim\u00e9nez-Hoyos, Qiming Sun, and Garnet K.-L. Chan, \u201cA Practical Guide to Density Matrix Embedding Theory in Quantum Chemistry\u201d, J. Chem. Theory Comput., 12, pp.\u00a02706\u20132719 (2016).","date":"2022-05-19 16:08:03","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\": 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\": 0.45012834668159485, \"perplexity\": 5540.495038842377}, \"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-2022-21\/segments\/1652662529538.2\/warc\/CC-MAIN-20220519141152-20220519171152-00477.warc.gz\"}"} | null | null |
Slovenska Karitas je socialno-karitativna organizacija rimskokatoliške cerkve v Sloveniji.
Odlikovanja in nagrade
Leta 1994 je organizacija prejela častni znak svobode Republike Slovenije z naslednjo utemeljitvijo: »za človekoljubno dejavnost pri reševanju človeških stisk«.
Viri in opombe
Glej tudi
Seznam prejemnikov častnega znaka svobode Republike Slovenije
Zunanje povezave
Uradna spletna stran
Rimskokatoliška cerkev v Sloveniji
Nosilci častnega znaka svobode Republike Slovenije
Slovenske humanitarne organizacije | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,891 |
Q: How many tab that be created in facebook page using graph api I have created one facebook page.I am developing an facebook application with asp.net 2.0 using C#.I wanted to add one asp page(suppose fb.asp) content to my facebook page. I have one iframe application in facebook,in its convas url given that fb page url.By using graph api I am able to add my application to my page(means content on of convas url to my facebook page).It creates one tab on page.Suppose I wanted to add diffrent tab on my one facebook page then what is concept behind it.Shall I create diffrent application for diffrent tab or one application is enough for creating diffrent tab.How many tab can I get on my facebook page.
suppose
Facebook page :x
pageid of fbpage :xx
application name :y
application id :yy
asp page that added as convas url for application y :sample.asp
convasurl: http://IpAddress//foldername/foldername/sample.asp
Graph url
https://graph.facebook.com/pageid/tabs?app_id=your application id&method=POST &access_token=page access token.
A: I assume what you're asking is can one Facebook tab application have more than one page and for each page to show up with its own tab marker in the left-hand column.
I don't believe that is the case (I believe you can only have one tab link per application), though if you have internal navigation within your application, you can certainly have several pages under the same "tab".
But I may have misunderstood your question.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,467 |
Q: Call command of web service from command line of Python I do the following Python commands:
import urllib
data = urllib.urlencode({"contains":"my_function"})
u = urllib.urlopen("http://myservername:1000/myfolder/?%s" % data)
u.read()
Then I get from that read command a lot of lines with HTML tags and one of the strings is of my interest. It looks like this:
...... onClick='doCommand("my_function","51267", $("ttt27222").value); $("ttt27222").value="";' >Apply
This is what I want to do from command line of Python using urllib.
Please let me know how to build urllib statement in order to call this my_function function passing it two parameters: 51267 and soem number for value.
Thank you
A: doCommand() seems like a javascript function. urllib doesn't execute javascript. You could use selenium webdriver, ghost.py to emulate web browser (to execute javascript in the context of the web page).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,406 |
Q: Resultado do sql no spring boot Boa tarde,
Preciso de uma ajuda para fazer um endpoint no Spring Boot, no banco de dados (MySQL), eu tenho a seguinte query:
SELECT U.NOME, P.DESCRICAO FROM USUARIO_PERMISSAO UP
JOIN USUARIO U ON UP.ID_USUARIO = U.ID
JOIN PERMISSAO P ON UP.ID_PERMISSAO = P.ID
A tabela Usuario_Permissão, ela é criada pelo banco, ManyToMany da tabela usuário e permissão, gostaria de saber como faço um endpoint que posso trazer o resultado.
Como não tenho essa "model", não estou conseguindo fazer o retorno do repository.
Estou tentando fazer assim:
@Query(nativeQuery = true, value = "SELECT U.NOME, P.DESCRICAO FROM USUARIO_PERMISSAO UP JOIN USUARIO U ON UP.ID_USUARIO = U.ID JOIN PERMISSAO P ON UP.ID_PERMISSAO = P.ID")
Object findByUsuarioPermissao();
@GetMapping
public Object findUsuarioPermissao() {
return usuarioRepository.findByUsuarioPermissao();
}
Porém na hora de testar está aparecendo esse erro:
"message": "query did not return a unique result: 20; nested exception is javax.persistence.NonUniqueResultException: query did not return a unique result: 20"
A: O erro é porque o método findByUsuarioPermissao espera o retorno de um objeto apenas e sua consulta está retornando mais de um objeto.
O ideal seria mudar para List<Object> ou limitar sua query para trazer apenas um resultado.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,898 |
{"url":"https:\/\/www.physicsforums.com\/threads\/correction-to-the-eigenvalue-doubly-degenerate-case.837893\/","text":"# Correction to the Eigenvalue -- doubly degenerate case\n\n1. Oct 15, 2015\n\n### Imperatore\n\nDetermine the corerctions to the eigenvalue in the first approximation and the correct functions in the zeroth approximation, for a doubly degenerate level.\n\nThe solution:\nEquation $$\\left| V_{nn'}-E^{(1)}\\delta_{nn'}\\right|=0$$ has here the form\n\n$$\\left|\\begin{array}{ccc}V_{11}-E^{(1)}&V_{21}\\\\V_{12}&V_{22}-E^{(1)}\\end{array}\\right|=0$$\n\nSolving, we find:\n\n$$E^{(1)}=\\frac{1}{2}\\left[ V_{11}+V_{22} \\pm h\\omega^{(1)}\\right]$$\n\nSolving also equation $$\\sum_{n'}(V-{nn'}-E^{(1)}\\delta_{nn'})c_{n'}^{(0)}$$ with these values of $$E^{(1)}$$, we obtain for the coefficients in the correct normalized function in the zeroth approximation , $$\\psi^{(0)}=c_{1}^{(0)}\\psi_{1}^{(0)}+c_{2}^{(0)}\\psi_{2)^{(0)}$$\n\n$$c_{1}^{(0)}=\\left\\{ \\frac{V _{12} }{2\\left| V _{12} \\right| } \\left[ 1 \\pm \\frac{V _{11}-V _{22} }{h\\omega ^{(1)} } \\right] \\right\\} ^{ \\frac{1}{2} }$$\n\n$$c_{2}^{(0)}= \\pm \\left\\{ \\frac{V _{21} }{2\\left| V _{12} \\right| } \\left[ 1 \\mp \\frac{V _{11}-V _{22} }{h\\omega ^{(1)} } \\right] \\right\\} ^{ \\frac{1}{2} }$$\n\nThe first equation I know, the task is just to solve the problem $$(V _{11}-E ^{(1)})(V _{22}-E ^{(1)})-V _{21}V _{12}=0$$ ut why there is$$\\left| V _{12} \\right| ^{2}$$ factor in the final formula ? What it represents ? And the most important, how to derive the formulas for these coefficients $$c_{1}^{(0)}$$$$c_{2}^{(0)}$$ ?\n\n2. Oct 20, 2015","date":"2018-03-22 07:24:35","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\": 0.9430046677589417, \"perplexity\": 734.5544312065049}, \"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-2018-13\/segments\/1521257647777.59\/warc\/CC-MAIN-20180322053608-20180322073608-00003.warc.gz\"}"} | null | null |
{"url":"http:\/\/mathematica.stackexchange.com\/questions\/45171\/locate-blow-up-in-ndsolve-with-whenevent","text":"# Locate Blow-up in NDSolve with Whenevent\n\nAt some point I get this error:\n\nWarning: scaled local spatial error estimate of 32773.0090044258 at\nt = 987.7919550194803 in the direction of independent variable x is much greater than the\nprescribed error tolerance. Grid spacing with 1001 points may be too large to achieve the\ndesired accuracy or precision.\n\n\n(etc.)\n\nHow do I track this time with WhenEvent? How is this scaled local spatial error kept in NDSolve?\n\n-\n\u2013\u00a0Michael E2 Apr 1 '14 at 16:31","date":"2016-05-03 01:16:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 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\": 0.7476303577423096, \"perplexity\": 2346.427308645435}, \"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-2016-18\/segments\/1461860118321.95\/warc\/CC-MAIN-20160428161518-00088-ip-10-239-7-51.ec2.internal.warc.gz\"}"} | null | null |
Q: PHP: Need loop to alternate between returned posts I have a an array of posts that are returned by doing three queries.
3 posts from the blog where posts are NOT in 'In the Media' or 'Insights',
3 from the blog where posts are in 'In the Media'
3 from the blog where posts are in 'Insights'.
Here's what I have for that. I don't think it's the most elegant solution:
<? $args = array(
'post_type' => 'post',
'posts_per_page' => 3,
'category__not_in' => array( 268, 269 )
);
$homePosts = new WP_Query($args);
$args = array(
'post_type' => 'post',
'category_name' => 'in-the-media',
'posts_per_page' => 3
);
$inthemediaPosts = new WP_Query($args);
$args = array(
'post_type' => 'post',
'category_name' => 'bt-insights',
'posts_per_page' => 3
);
$insightsPosts = new WP_Query($args);
$allqueries = array($homePosts,$inthemediaPosts,$insightsPosts);
foreach ($allqueries as $myquery) {
while ($myquery->have_posts()) : $myquery->the_post(); ?>
Currently, that loops through 3 homeposts, then 3 inthemedia posts, then 3 bt-insight posts.
What I need is for the loop to go through 1 homepost, 1 inthemedia post, 1 bt-insight post, then 1 homepost, 1 inthemediea post... etc repeat.
Hope that makes sense. Suggestions?
A: while($allqueries[0]->have_posts() || $allqueries[1]->have_posts() || $allqueries[2]->have_posts()) {
if ($allqueries[0]->have_posts()) $allqueries[0]->the_post();
if ($allqueries[1]->have_posts()) $allqueries[1]->the_post();
if ($allqueries[2]->have_posts()) $allqueries[2]->the_post();
}
A: What if you remove the allqueries stuff and after:
$insightsPosts = new WP_Query($args);
Just use this instead.
for ($i = 0; $i < 3; $i++) {
if ($homePosts->post_count > $i)
echo $homePosts->posts[$i]->post_title;
if ($inthemediaPosts->post_count > $i)
echo $inthemediaPosts->posts[$i]->post_title;
if ($insightsPosts->post_count > $i)
echo $insightsPosts->posts[$i]->post_title;
}
That should just print out the title of the post and you can use the other fields as needed. This could be moved to a function to avoid duplicating the logic as well.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,552 |
Leni Robredo, BBM Supporters Cross Paths in Caravans in Camarines Sur
by Sandy Ghaz
Date: November 22, 2021
in: Election 2022
Caravans To Show Support for Vice Pres. Leni Robredo, Former Sen. Bongbong Marcos Jr. Held in Pili, Camarines Sur
LENI ROBREDO, BBM – The supporters of the two presidential aspirants crossed paths in caravans held in Pili, Camarines Sur.
Although Election 2022 is still around six (6) months away and the campaign period has yet to officially start, there are already a lot of events held across the nation to show support to certain political candidates. Respective caravans were held to show support to presidential aspirants Vice Pres. Leni Robredo and former Sen. Bongbong Marcos Jr. by their supporters.
Former Sen. Bongbong Marcos Jr. or also called by his supporters "BBM" is running for president under Partido Federal ng Pilipinas (PFP). His party tapped Davao City Mayor Sara Duterte-Carpio as their vice-presidential bet. Their fans and supporters have long called for the BBM-Sara tandem.
Undeniably, the presidential bid of the former Senator triggered the resurfacing of the issues surrounding the regime of his late father, their family's alleged ill-gotten wealth, and the personal questions on his qualification for the presidency. Amid five (5) petitions that were filed against his bid, Marcos Jr. is determined to pursue running for president in 2022 and his supporters are hopeful that he will make the country "great again".
Photo lifted from Philippine Star
Vice Pres. Leni Robredo is the presidential nominee of 1SAMBAYAN. She filed her Certificate of Candidacy (COC) for president as an independent candidate.
The Vice President said that deciding to run was a hard decision to make admitting that they don't have enough resources for the campaign. Meanwhile, her decision was more affirmed by the support shown to her by her supporters who held different activities to show support to her.
Caravans to show support to her were held in several cities in the Philippines. There were also feeding events featuring "lugaw" – which is linked to the Vice President.
Caravans were also held for other presidential bets by their supporters. Recently, the respective caravans to show support for Vice Pres. Leni Robredo and BBM crossed paths in Pili, Camarines Sur. Here's a photo posted on Facebook:
Facebook/Pasada Rinconada Camarines Sur
Thank you for visiting Philnews.ph. You may express your thoughts and reactions below through the comment section. You may also follow us on Facebook, on Twitter, and subscribe to our YouTube channel Philnews Ph.
READ ALSO: Bongbong Marcos Camp Respectfully Prays the Petition to be Dismissed
comment(s) for this post "Leni Robredo, BBM Supporters Cross Paths in Caravans in Camarines Sur". Tell us what you think abut this post by leaving your comments below.
Tagged as: bbm, Bongbong Marcos Jr, Caravan Bongbong Marcos, Caravan Leni Robredo, Election 2022, Leni Bongbong Caravan, Leni Robredo | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,258 |
{"url":"http:\/\/math.stackexchange.com\/questions\/66245\/re-writing-a-logarithm-to-a-power\/66250","text":"# Re-writing a logarithm to a power\n\nGiven:\n\n$$(4\\ln x)^2$$\n\nIs this simplified to $8\\ln x$, (multiplying the expression by 2),\n$32\\ln x$, (square $4$ ($16$), then $\\ln x$ ($2\\ln x$) and combine again),\nor something else?\n\nJust to be sure, the equation I was solving for was $e^\\sqrt t$ = $x^4$ for $t$.\nMy thought process is:\n\n$\\ln(x^4)$ = $\\sqrt t$\n$4\\ln x$ = $\\sqrt t$\n$(4\\ln x)^2$ = $t$\n\n-\nIf the logarithm itself is squared, that's as simple as it can get. If what you had instead was $\\ln x^2$, then something can be done... \u2013\u00a0 \uff2a. \uff2d. Sep 21 '11 at 1:40\nSo is that already in simplest form? (Also, would you mind checking the edit I added to the bottom, just to ensure that I'm doing this right. I'm attempting to remember how to use logorithms at this point, so I'm still a beginner). \u2013\u00a0 Mike Gates Sep 21 '11 at 1:42\nThe expression $(\\ln x)^2$ is the same as $\\ln x \\cdot \\ln x$. The expression $2\\ln x$ is the same as $\\ln x + \\ln x$. Multiplying two numbers and adding two numbers should, in general, give different results. \u2013\u00a0 Austin Mohr Sep 21 '11 at 1:48\nWell done. One might want to rewrite the answer as $16(\\ln x)^2$. \u2013\u00a0 Andr\u00e9 Nicolas Sep 21 '11 at 5:26\nIf you really want to get rid of the exponent two you make things more complicated: $(\\ln x)^2 = (\\ln x)(\\ln x) = \\ln(x^{\\ln x})$... \u2013\u00a0 Dirk Sep 21 '11 at 6:19\n\nIt might be helpful to \u201ctake out\u201d the logarithm for a second so you don\u2019t get hung up on log properties. Let\u2019s take $a = 4$ and $b = \\ln x$; then\n$$(4 \\ln x)^2 = (ab)^2 = a^2 \\cdot b^2 = 4^2 \\cdot (\\ln x)^2$$\nIt\u2019s pretty clear now what the coefficient is going to be, and as J.M. pointed out above, there\u2019s nothing more to be done with a term like $(\\ln x)^2$.\nAs for the underlying question, you can substitute $\\sqrt{t}$ into your expression easily enough; then just apply your exponential\/logarithmic rules and check that the two sides are equal.","date":"2015-01-31 22:40:59","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\": 0.9209457039833069, \"perplexity\": 487.6174091204956}, \"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-2015-06\/segments\/1422115861162.19\/warc\/CC-MAIN-20150124161101-00038-ip-10-180-212-252.ec2.internal.warc.gz\"}"} | null | null |
{"url":"http:\/\/vuthy.ligeracademyblog.org\/2018\/12\/","text":"## Long and Synthesis Division\n\nAfter many months of SAT preparation and the first SAT exam was finished, our Math class tilts toward precalculus. We studied the function\u2019s behavior, transformation, and shapes. Many times, we find zeros of polynomials through couple strategies such as Descartes\u2019s rule of Signs, Upper and Lower Bounds for zero, list of possible zeros (factor of constant term\/factor of leading coefficient), factoring and long division and synthetic division.\n\nI found out that long division and synthetic division to solve for zeros is really essential, so I want to show how to use them.\n\n1. Long Division\n\nIt looks almost the same with normal division we did in grade 2. However, complexity but the same element.\n\n2.\u00a0 Synthetic division\n\nThe same problem as above, the synthetic division has a different way but result in the same answer.\n\n## Chemistry Molecular shape\n\nAs class progress, our chemistry knowledge is deeper. Each lesson builds upon each other. For instance, after learning the Lewis dot structures, we have gone farther about molecular bonding shape of a compound and its polarity.","date":"2021-01-20 20:17:18","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\": 0.8055037260055542, \"perplexity\": 1719.249028870165}, \"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-2021-04\/segments\/1610703521987.71\/warc\/CC-MAIN-20210120182259-20210120212259-00493.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.yengmillerchang.com\/post\/lm-theory-in-practice\/","text":"# Linear Regression: Theory in Practice\n\nViews and opinions expressed are solely my own.\n\n## Introduction\n\nIn this post, we will be building from the linear regression theory post and applying these concepts to a practical problem.\n\nMy interest in writing this post is in outlining the complexities of doing a statistical analysis that are often omitted from textbooks, especially for a procedure that may appear to be as simple as fitting a line through points (linear regression). Statistics and machine learning textbooks often give the appearance that statistical choices may be based on a sequence of binary \u201cthis-is-true\u201d or \u201cthis-is-not-true\u201d decisions.\n\n## Simulating the Setting\n\nAs mentioned in the previous post, we assume\n\n$Y_i = \\beta_0 + \\beta_1X_i + \\epsilon_i = \\text{intercept } + \\text{slope }\\cdot X_i + \\unicode{x201C}\\text{noise\/error}\\unicode{x201D}$\n\nwith $$\\mathbb{E}[\\epsilon_i \\mid X_i] = 0$$. Notice it follows immediately by double expectation that $$\\mathbb{E}[\\epsilon_i] = 0$$. Here is an example of such data. For convenience, we assume that the $$\\epsilon_i$$ are normally distributed with mean $$0$$ and variance $$1$$ in our simulation.\n\nlibrary(ggplot2)\n\nset.seed(45)\n\n# generate a data frame of n x-y pairs based on\n# the linearity + noise assumption\ngenerate_df <- function(n, beta_0, beta_1,\nerror_function, ...) {\n# Generate random x values between 0 and 20\nx <- runif(n = n, min = 0, max = 20)\n\n# Compute y_i\ny <- beta_0 + beta_1 * x + error_function(n = n, ...)\n\n# Create data frame\nreturn(data.frame(x = x, y = y))\n}\n\n# Set beta_0 to 0.5, beta_1 to 1\nbeta_0 <- 0.5\nbeta_1 <- 1\n\ndf <- generate_df(n = 200,\nbeta_0 = beta_0,\nbeta_1 = beta_1,\nerror_function = rnorm,\nmean = 0,\nsd = 1)\n\n# Define function for the true process, without the error term\nf <- function(x, beta_0, beta_1) {\nreturn(beta_0 + beta_1 * x)\n}\n\n# Plot the graph\nggplot(df, aes(x = x, y = y)) +\ngeom_point() + # dots\n# the true process without the noise\nstat_function(fun = f,\nargs = list(beta_0 = beta_0, beta_1 = beta_1),\ncolor = \"forestgreen\",\nsize = 2) +\n# black-and-white theme\ntheme_bw() +\n# blank panel grid\ntheme(panel.grid = element_blank())\n\nWhat the visualization is portraying above is that the true $$Y_i$$ values are a linear equation (in green) plus some \u201cnoise\u201d whose mean is $$0$$.\n\n## Application to Actual Data\n\nWe work with the Advertising.csv file from the second edition of An Introduction to Statistical Learning:\n\ndf <- read.csv(\"https:\/\/www.statlearning.com\/s\/Advertising.csv\",\nrow.names = 1) # row 1 contains the row labels\n\nhead(df) # show first six rows\n## TV radio newspaper sales\n## 1 230.1 37.8 69.2 22.1\n## 2 44.5 39.3 45.1 10.4\n## 3 17.2 45.9 69.3 9.3\n## 4 151.5 41.3 58.5 18.5\n## 5 180.8 10.8 58.4 12.9\n## 6 8.7 48.9 75.0 7.2\n\nThese data consist of $$n = 200$$ observations with $$p = 4$$ columns:\n\n\u2022 sales in thousands of units for a particular product, as well as\n\u2022 TV, radio, and newspaper advertising budgets in thousands of dollars.\n\nSuppose we treat $$Y_i$$ as sales and $$X_i$$ as TV. Let\u2019s plot these data and see how they look:\n\n# Plot the graph\nggplot(df, aes(x = TV, y = sales)) +\ngeom_point() + # dots\n# black-and-white theme\ntheme_bw() +\n# blank panel grid\ntheme(panel.grid = element_blank()) +\n# x-scale: from 0 to 300, ticks every 25\nscale_x_continuous(limits = c(0, 300),\nbreaks = seq(0, 300, 25)) +\n# y-scale: from 0 to 30, ticks every 5\nscale_y_continuous(limits = c(0, 30),\nbreaks = seq(0, 30, 5))\n\nFor simple linear regression, we would then assume that for each observation $$i = 1, \\dots, 200$$ that the following relationship holds for some intercept $$\\beta_0$$, slope term $$\\beta_1$$, and noise term $$\\epsilon_i$$ (with $$\\mathbb{E}[\\epsilon_i \\mid \\text{TV}_i] = 0$$) for each observation: $\\begin{equation*} \\text{Sales}_i = \\beta_0 + \\beta_1 \\cdot \\text{TV}_i + \\epsilon_i\\text{.} \\end{equation*}$\n\n### Is this assumption true?\n\nThe reality is that, unless we are in control of how the data are generated and know that the data may be specified by the process above, we cannot guarantee that the $\\begin{equation*} \\text{Sales}_i = \\beta_0 + \\beta_1 \\cdot \\text{TV}_i + \\epsilon_i \\end{equation*}$ assumption is true. That is, given a bunch of TV values, unless we know for a fact that Sales were based on the form given above (like in our simulation we did earlier in this post), we cannot guarantee this assumption holds.\n\nSo the question of \u201cis this assumption true?\u201d turns into\n\n### Is this assumption good enough?\n\nThis question is a much more difficult question to answer than the prior one, and is often ignored.\n\nTwo things I do not suggest for answering this question, though, include:\n\n\u2022 Hypothesis testing for distributional fit or finding evidence of the coefficient $$\\beta_1$$ being not equal to $$0$$ (such as through a $$t$$-test): such procedures encourage binary, \u201cstrict-cutoff\u201d-type thinking which may give misleading results. Additionally, such hypothesis tests assume that the $$\\text{Sales}_i = \\beta_0 + \\beta_1 \\cdot \\text{TV}_i + \\epsilon_i$$ assumption is true to begin with.\n\u2022 Relying solely on graphical procedures to test these assumptions: there are many arbitrary decisions that go into making a graph that could influence how you view the data (e.g., the axes bounds, ticks, etc.).\n\nThe question of interest matters a lot than many textbooks suggest. If someone poses to me a question similar to \u201chow does TV advertising budget influence sales?\u201d I would be inclined to assume the model $$\\text{Sales}_i = \\beta_0 + \\beta_1 \\cdot \\text{TV}_i + \\epsilon_i$$ just to get a \u201cgood-enough\u201d result to start with, and if necessary, I can make it more complex from there. In such cases, one prioritizes intepretability over precision.\n\nQuestions that are more detailed than \u201chow does TV advertising budget influence sales?\u201d - particularly if there is interest in a sales prediction involved given a TV budget - may require more sophisticated tools than linear regression, such as local regression, with some cross validation thrown in. However, as one increases the level of sophistication of statistical tools, there is usually more of a problem of intepretability to general audiences. Back when I taught introductory data science, I referred to this phenomena as the explainability vs.\u00a0complexity tradeoff.1\n\nFor the purposes of this post, we will assume that the model $$\\text{Sales}_i = \\beta_0 + \\beta_1 \\cdot \\text{TV}_i + \\epsilon_i$$ holds with $$\\mathbb{E}[\\epsilon_i \\mid \\text{TV}_i] = 0$$.\n\n### Executing the regression procedure, and interpretation\n\nIf we assume $$\\text{Sales}_i = \\beta_0 + \\beta_1 \\cdot \\text{TV}_i + \\epsilon_i$$ holds with $$\\mathbb{E}[\\epsilon_i \\mid \\text{TV}_i] = 0$$, by the Gauss-Markov theorem, the least-squares estimators of $$\\beta_0$$ and $$\\beta_1$$ are their corresponding \u201cbest\u201d estimators. In R, these may be calculated using the lm (linear model) function:\n\n# execute the least-squares procedure\nmodel <- lm(sales ~ TV, data = df)\n\n# return summary stats\nsummary(model)\n##\n## Call:\n## lm(formula = sales ~ TV, data = df)\n##\n## Residuals:\n## Min 1Q Median 3Q Max\n## -8.3860 -1.9545 -0.1913 2.0671 7.2124\n##\n## Coefficients:\n## Estimate Std. Error t value Pr(>|t|)\n## (Intercept) 7.032594 0.457843 15.36 <2e-16 ***\n## TV 0.047537 0.002691 17.67 <2e-16 ***\n## ---\n## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n##\n## Residual standard error: 3.259 on 198 degrees of freedom\n## Multiple R-squared: 0.6119, Adjusted R-squared: 0.6099\n## F-statistic: 312.1 on 1 and 198 DF, p-value: < 2.2e-16\n\nThe least squares estimators of $$\\beta_0$$ and $$\\beta_1$$ are $$\\hat{\\beta}_0 = 7.032594$$ and $$\\hat{\\beta}_1 = 0.047537$$ respectively. These may be interpreted as follows:\n\n\u2022 A product with zero (thousands of) dollars of TV advertising budget sells roughly 7,033 ($$1000 \\cdot \\hat{\\beta}_0$$) units of the product. This interpretation is based on the value of $$\\beta_0$$, and recalling that both $$\\text{TV}_i$$ and $$\\text{Sales}_i$$ are measured in thousands.\n\u2022 Adding one thousand dollars to the TV advertising budget for a product is associated with a \\$47.54 ($$1000 \\cdot \\hat{\\beta}_1$$) increase in sales.\n\nIt is important to note that these interpretations rely on all of the following assumptions holding:\n\n\u2022 $$\\text{Sales}_i = \\beta_0 + \\beta_1 \\cdot \\text{TV}_i + \\epsilon_i$$, and\n\u2022 $$\\mathbb{E}[\\epsilon_i \\mid \\text{TV}_i] = 0$$.\n\nThe first assumption above is especially important, in that no other variables other than $$\\text{TV}_i$$ are included in the determination of $$\\text{Sales}_i$$.\n\n## Some other mathematical concerns\n\nAs mentioned in the prior linear regression theory post, if we wish to calculate the variance of the least-squares estimators of $$\\beta_0$$ and $$\\beta_1$$, we must also assume that $$\\text{Var}[\\epsilon_i \\mid \\text{TV}_i] = \\sigma^2$$, which implies that $$\\text{Var}[\\text{Sales}_i \\mid \\text{TV}_i] = \\sigma^2$$. This condition is also known as homoskedasticity. In simple terms, homoskedasticity assumes that the variability of the $$\\text{Sales}_i$$ is constant regardless of the value of $$\\text{TV}_i$$ we have. When this condition does not hold, the model is said to exhibit heteroskedasticity, which is given emphasis in econometrics more often than statistics.2 I haven\u2019t had much of a chance to dive into this world, but may cover this material in a future post.\n\nFrom a naive visual inspection, one should observe that the $$\\text{Sales}_i$$ values seem to be more spread out as the $$\\text{TV}_i$$ values increase, which would suggest that we have heteroskedasticity. The variability of $$\\text{Sales}_i$$ values seems to increase as $$\\text{TV}_i$$ increases, roughly approximated by the red lines and arrows below.\n\nIf we desire, we could assume the model $$\\text{Sales}_i = \\beta_0 + \\beta_1 \\cdot \\text{TV}_i + \\epsilon_i$$ holds with the variance of $$\\epsilon_i$$ being proportional to $$X_i$$; i.e., $$\\text{Var}[\\epsilon_i \\mid \\text{TV}_i] = \\sigma^2 \\cdot \\text{TV}_i$$. This is a more complicated version of simple linear regression than we\u2019ve discussed so far, known as the Aitken Model or Generalized Least Squares. I may write a post on this topic in the future.\n\n## Conclusion\n\nI hope this posts illustrates how complex a simple procedure such as univariate linear regression is, with regard to the thought process that has to go into validating assumptions and making tradeoffs between complexity and intepretability. Statistical education needs to make such thought processes more explicit.\n\n## References\n\nJames, G., Witten, D., Hastie, T., & Tibshirani, R. (2021). An Introduction to Statistical Learning (2nd ed.). New York, NY: Springer Science+Business Media.\n\n1. Artificial Intelligence is trying to address this issue through explainable AI, but I am a skeptic about the long-term implications of this movement being successful.\u21a9\ufe0e","date":"2022-05-26 14:49:48","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\": 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\": 0.7677074074745178, \"perplexity\": 1205.464369517927}, \"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-2022-21\/segments\/1652662606992.69\/warc\/CC-MAIN-20220526131456-20220526161456-00763.warc.gz\"}"} | null | null |
In 1919 America was still recovering from from the Great World War. Price inflation and the cost of living had increased far beyond wages. Men back from the war flooded the labor market which further diminished workers' earning power.
During this year, one fifth of the country's workers would strike. The year started with New York's harbor workers striking in January, followed by dressmakers. In February news headlines reported a "prelude to revolution" when a general strike in Seattle closed all business February 6 to 11th.
Bombs were mailed to the mayor of Seattle who broke the strike, and in April, forty more mail bombs were found en route to other public leaders for May Day. With the backdrop of apparently spreading communism, many patriotic Americans believed they were on the verge of a workers revolution.
In Massachusetts, textile workers in Lawrence walked out in protest of six day, nine hour work schedules. Boston telephone operators interrupted much of New England's phone service in an April, and in July Boston's elevated train workers went off the job in their own protest. Boston's political and business leaders could see this national trend of strikes disabling their own businesses and communities, and they were alarmed.
There is no doubt that Boston's police had grievances, which they expressed as early as 1917. New officer pay had not risen in sixty years, since 1857 when new recruits received two dollars daily. Officers worked seven days per week, with a day off every other week during which they couldn't leave town without special permission. Depending on duty, officers worked between 73 and 98 hours weekly, and were required to sleep in infested station houses kept in deplorable condition.
Since 1885, the Boston police were under the command of a commissioner appointed by the state governor. Though Boston's mayor controlled their budget, their operation and how they used the budget was controlled by a commissioner appointed by the governor. This placed the mayor, Andrew Peters in a difficult position. His city was protected by a force not in his control. When they would succeed the state took credit, when there were problems he was closet to them.
There was also an ethnic overlay. Protestants Yankees sought to control the Irish-Catholic rank and file of the Boston Police Department. This made the dispute about more than wages or work conditions, it quickly broke along lines of ethnicity.
By June of 1919 the grievances made by the police had not been addressed. They turned to the American Federation of Labor to consider unionization. Police Commissioner Curtis was outspoken in his condemnation of the movement to unionize.
In August they would be granted a union charter, which Commissioner Curtis would oppose on the grounds that police were not "an employee, but a state officer." Mayor Peters was unreachable on an extended vacation in Maine, but Governor Coolidge and Attorney General Albert Pillsbury put forward legislation to make unionization illegal for public employees. Pillsbury would note that the "organized work man has taken us by the throat and has us at his mercy". The lines of "us" and "them" were drawn quickly.
From this point on, state officials focused on the legitimacy of public employees unionizing rather than the validity of the officers complains. On August 20th commissioner Curtis suspended eight of the leading police union organizers, followed soon by another ten. The rank and file were ordered to turn in their nightsticks, and Curtis began to organize volunteer police substitutes.
When mayor Peters returned from vacation he made conciliatory statements, and organized a commission headed by prominent banker James Jackson Storrow. The commission came to the expected conclusion, that though wages and working conditions required change, it would be inconsistent with public interest to negotiate with a union, or have the police obliged to any non-public organization. The majority need for public safety outweighed the officers asserted right to collective bargaining.
The police were emotionally committed to the union as the only effective means of making progress. The vote to strike was within five votes of unanimous, and was seen in Massachusetts as the climax to a year of violence and threats.
Senator Henry Cabot Lodge described the strike as "the first step to sovietizing the country". Governor Coolidge suggested that the city council raise wages and improve working conditions. Still, Coolidge's commissioner held firm to his position, and recruited about 200 Harvard athletes and business men to step in during the expected strike.
Mayor Peters requested that Coolidge dispatch the state guard as a first reaction to the strike. Coolidge sided with his police commissioner who advised that volunteers could fill the void, and that troops should only be sent if needed.
On the evening of September 9th the strike took place as planned. Nearly three quarters of Boston's police left their jobs. As residents absorbed the reality of the absence of police, the mischievous among them took the opportunity to engage in petty crimes. There was gambling in public, harassing officers: both those on strike who were now without badge and gun, as well as those who remained as strike breakers in the greatly weakened force.
By 8pm a crowd estimated at 10,000 gathered, as pictured here, in Scollay Square. Soon a cigar store window was broken and emptied. This was followed by a frenzy of looting, gambling and mayhem downtown and in South Boston until after 1am.
That morning political positioning ensued. Mayor Peters issued a press release saying he was not to blame, and he called out the state guard. He used an emergency clause to take control of the police whenever "tumult, riot, and violent disturbance" happened within the city. Coolidge reacted by issuing a statement that as Governor the guard would be under his control, and that Coolidge not Peters would save Boston.
The President of Harvard, Lawrence Lowell called on more students to volunteer, as did Boston's businessmen. Crowds waited outside police stations to attack the volunteers. Cries were heard of "kill the cops". The Commander of Station 6 in South Boston kept his Harvard volunteers in the station to protect their lives.
At Scollay Square there were sporadic confrontations between the replacement police and crowd, resulting in several Harvard students being cornered. When the first troop of cavalry arrived they had to intervene to rescue groups of cornered police. Several guardsmen were injured by thrown rocks, but eventually the threat of live ammunition and horsemen with swords pushed the crowd from Scollay Square.
Boston a hive of military activity. At the old armory near the Park Plaza hotel mobile units with machine guns set up headquarters. The guard forces kill five residents, while three more would by killed by civilians. Over twenty people sustained serious wounds.
The next morning the LA Times wrote "...no man's house, no man's wife, no man's children will be safe if the police force in unionized and made subject to the orders of Red Unionite bosses."
Mayor Peters feared that a general strike such as the one which closed Seattle would follow with the support of other unions and public employees. With order restored, he met with union leaders to seek compromise. Governor Coolidge could afford to take a firmer stance, as ultimately he would not have to lead the city through a general strike. Further he had a supporter, "Diamond" Jim Timilty sitting inside the Central Labor Organization who secretly promised a general strike would not be made.
When the Police Union asked that the officers be reinstated so they could protect the city while a new contract was negotiated, Coolidge rebuffed by refusing and stating "there is no right to strike against the public safety by anybody, anywhere, any time."
Labor was plentiful, by mid-December Commissioner Curtis had hired an entirely new police force at increased wages and with better working conditions. The state guard was was able to return to their homes, and no striking officer returned to the Boston police force.
Governor Coolidge was easily reelected on November 4th, 1919. A year later he would become Vice-President of the United States, and following the death of President Warren Harding he became President on August 2, 1923. Mayor Peters would be defeated in his next election by his political rival James Michael Curley, who had preceded him as Mayor. | {
"redpajama_set_name": "RedPajamaC4"
} | 794 |
Q: What is the Twisted Essence of Suffering used for? 2 of these was dropped after fighting Andariel in Hell Difficulty, they can't be traded or sold, what are they used for?
Presumably some Hordadric Cube recipe?
A: You are correct that it is a horadric cube recipe. It creates a Token of Absolution that allows you to respec. a character.
Added in Patch 1.13, the Token of Absolution is a consumable item that
grants the character one stat/skill reset. This works in the same way
as the reward from the Den of Evil quest, but allows a character
infinite respecs, provided that they are willing to gather the
reagents each time they desire a respec. The Token of Absolution is
created in the Horadric Cube by cubing together four different
Essences that drop infrequently from the act bosses in Hell
difficulty.
The Essences include:
*
*Twisted Essence of Suffering (dropped by Andariel and Duriel)
*Charged Essence of Hatred (dropped by Mephisto)
*Burning Essence of Terror (dropped by Diablo)
*Festering Essence of Destruction (dropped by Baal).
They create: Token of Absolution
source: Diablo 2 Wikia
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,489 |
$.backstretch([
"/static/images/lockscreen/1.jpg",
"/static/images/lockscreen/2.jpg",
"/static/images/lockscreen/3.jpg",
"/static/images/lockscreen/4.jpg",
"/static/images/lockscreen/5.jpg",
"/static/images/lockscreen/6.jpg",
"/static/images/lockscreen/7.jpg",
"/static/images/lockscreen/8.jpg",
"/static/images/lockscreen/9.jpg",
"/static/images/lockscreen/10.jpg",
"/static/images/lockscreen/11.jpg",
"/static/images/lockscreen/12.jpg",
"/static/images/lockscreen/13.jpg",
], {
fade: 1000,
duration: 5000
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,584 |
Many cities in the U.S. have begun allowing electric scooters, and many other cities are considering legislation that would soon allow electric scooters. For example, electric scooter legislation was introduced in New York toward the end of 2018, and a recent report from KYW News Radio Philly reports that electric scooters could soon be available in Philadelphia if a new piece of legislation passes.
However, e-scooters (as they are commonly called) also pose significant safety risks. According to a Consumer Reports investigation, the dangers of electric scooters are real. E-scooter accidents are causing thousands of personal injuries across the country, and emergency departments are failing to accurately track them.
In some cases, patients require surgery for serious injuries. Many e-scooter users do not wear helmets because they are not available to rent with the scooters, and few people carry helmets with them. Wearing a helmet is one of the most important parts of safe scooter riding and electric scooter safety.
While some emergency departments are keeping track of e-scooter injuries, others are not. The two largest electric scooter companies, Bird and Lime, also indicated that they do not currently have the capacity to track all injuries associated with these products.
The key piece of information from the investigation is that without safe scooter riding, e-scooters result in far more injuries than previously assumed. This e-scooter statistic includes collisions with pedestrians and bicyclists, as well as accidents involving motorists.
Avoid using a scooter altogether if there is a lightning storm.
If you were injured in an e-scooter collision with a motor vehicle, you should speak with a personal injury lawyer about your case. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,953 |
A novelist, Kim Jeongseok is looking for his missing wife. A woman appears in front of him and suggests to find his wife. Meanwhile, there are someone to approach and ask Kim Jeongseok to find missing persons. They finds themselves becoming chasers at some point.
The Avian Kind is the road movie that Kim pursues his missing wife, is a story that different people tries to resolve unknown truth together as well. During the journey, they meet another persons and the new ones also go forward to huge unknown being hiding their goal secretly.
While watching the film, the audiences would get curious more and more. Where is the wife disappeared? Is she changed into other existence like the rumor? Is it really possible?
Shin Younshick has brought up diverse hidden desires in human being consistently since early his films. He creates a story of human's disappeared dream of becoming new being. It is the origin that human has dreamt from time immemorial, at the same time, it is the desire to destroy the human´s boundary, further, it is also the story of salvation to wish new existence.
It is interesting that The Avian Kind deriving from a part of The Russian Novel is genealogical experiment that one film produces another film´s motif. Accumulating the layers of many stories, Mise-en-Scene of the film capturing the border between human and nature comes to audiences beautifully, hurtfully and keenly.
Known for his literary words and directing, Director Shin Younshick has been flexible enough to direct both commercial and independent films and have The Russian Novel(2012) and Rough Play(2013) released in previous years. In Jeonju Digital Project 2014, Shin presents The Avian Kind, which depicts a novelist's journey to search for his missing wife. It is also a title of a novel that appeared in his previous work The Russian Novel, and he plans to continue to take a cue from his previous films. Similar to The Russian Novel, The Avian Kind shows Shin's signature style crossing the present and the past. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,521 |
{% extends "base.html" %}
{% block content %}
<form method="post">
<label for="password">Password:
<input type="password" name="password" value="Password">
</form>
{%- endblock %}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,043 |
{"url":"http:\/\/wwww.thermalfluidscentral.org\/encyclopedia\/index.php?title=Integral_governing_equations&diff=next&oldid=1944","text":"# Integral governing equations\n\n(Difference between revisions)\n Revision as of 00:59, 6 November 2009 (view source)\u2190 Older edit Revision as of 01:04, 6 November 2009 (view source) (Blanked the page)Newer edit \u2192 Line 1: Line 1: - The law of the conservation of mass dictates that mass may be neither created nor destroyed.\u00a0 For a control volume that contains only one phase, conservation of mass can be obtained by setting the general and specific property forms to $\\Phi = m$ and $\\phi = 1$ in eq. (2.3), i.e., -\n${\\left. {\\frac{{dm}}{{dt}}} \\right|_{system}} = \\frac{\\partial }{{\\partial t}}\\int_V {\\rho dV + \\int_A {\\rho ({{\\mathbf{V}}_{rel}} \\cdot {\\mathbf{n}})dA} } \\qquad \\qquad(1)$\n- - where the first term on the right hand side represents the time rate of change of the mass of the contents of the control volume, and the second term on the right hand side represents the net rate of mass flow through the control surface. The term $({{\\mathbf{V}}_{rel}} \\cdot {\\mathbf{n}})dA$ in the mass flow integral represents the product of the velocity component perpendicular to the control surface and differential area. This term is the volume flowrate through $dA$, and becomes the mass flowrate when multiplied by density, ''\u03c1''. - - Since the mass of a fixed-mass system is constant by definition, and the fixed-mass system contains only one phase, the resulting formulation of conservation of mass is - -\n$\\frac{\\partial }{{\\partial t}}\\int_V {\\rho dV + \\int_A {\\rho ({{\\mathbf{V}}_{rel}} \\cdot {\\mathbf{n}})dA} } = 0 \\qquad \\qquad(2)$\n$\\sum\\limits_{k = 1}^\\Pi {\\left[ {\\frac{\\partial }{{\\partial t}}\\int_{{V_k}(t)} {{\\rho _k}dV} + \\int_{{A_k}(t)} {{\\rho _k}({{\\mathbf{V}}_{k,rel}} \\cdot {{\\mathbf{n}}_k})dA} } \\right]} = 0 \\qquad \\qquad(3)$\n- - where $k$ denotes the $k$th phase in the multiphase system, and $\\Pi$ is the number of phases. - - ==References== - - Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Burlington, MA. - - ==Further Reading== - - ==External Links==","date":"2022-12-07 20:02:36","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\": 0.894409716129303, \"perplexity\": 742.0586678214862}, \"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-2022-49\/segments\/1669446711218.21\/warc\/CC-MAIN-20221207185519-20221207215519-00656.warc.gz\"}"} | null | null |
Two extraordinary football stories are set to duke it out for the premiership cup. Here's the Xs and Os that will shape the outcome.
Football is chaos, manifest in millions of actions and reactions across 120 odd minutes of kicks and handballs, marks and tackles.
But amidst it all there is some order, some signal which we can look to as a way of helping work out who holds the advantage.
Both West Coast and Collingwood are worthy, albeit perhaps surprising, grand finalists. Each has a compelling football tale to tell.
But once the ball is bounced the stories don't matter anymore; it's all about who can execute their game plan better and for longer, and who can resist their opponent doing the same.
Despite it being a grand final between two of the 'powerhouse' clubs – who combined hold about a third of the stored financial worth of the 18 clubs – the build-up has been more muted than the build-up to recent grand finals.
That is perhaps a product of each club's heft and status; West Coast played in a grand final three years ago, and Collingwood won its last flag in 2010.
While each club has a great story to tell, it isn't chasing a historic threepeat, breaking a decades-long hoodoo, continuing a modern rivalry or playing for its first premiership.
There has, however, been plenty of talk about our first factor: the venue.
Give me any argument about the MCG not being an advantage for a tenant club or not being a disadvantage for an interstate club and I'll show you a terrible argument. Home ground advantage is an indisputable concept in sport – everywhere – and it applies in this situation.
Every home team in every professional sport played on every continent performs better than every away team in every professional sport played on every continent on average over the long run. Indisputable.
The ABC, picking up on the themes of Footballistics and additional commentary on the matter by some of the contributors, wrote an excellent explainer piece earlier in the week that should change the mind of any hold out.
As the piece notes, what is up for debate is what causes the advantage to occur: home crowds, ground familiarity and travel are some of the common theories.
The MCG had been a challenging ground for the Eagles prior to 2018. West Coast had won just one of their five games at the 'G following the 2015 grand final loss, and that was a seven-point victory over Carlton. That form has possibly turned, albeit on a small sample: two wins in two games, including against the Pies in Round 17.
One theory is Perth Stadium is of similar shape to the MCG; not quite as round around the middle but it is certainly wider and shorter than Subiaco Oval.
And Collingwood hasn't necessarily been at their best on their home ground in 2018. The Pies have an 8-6 record with a percentage of 113.5 per cent; their record against top eight adversaries at the MCG is 1-6 with a percentage of 80 per cent flat. Maybe home ground advantage isn't all its cracked up to be for the Pies after all.
Grand final matchups have been a strategist's dream in recent years, and 2018 is no exception. In the blue and yellow corner we have the kick-mark West Coast Eagles, playing a game designed to counter the chaosball and smallball preferences sweeping the league.
It's not dissimilar to the way Adelaide tried to move the ball last year, albeit the Eagles are more content controlling the ball until they find a lane they like (where the Crows played more aggressively), and West Coast being a bit more defensively minded.
In the black and white corner we have the run-chase Collingwood Magpies, who play for territory and treat the ball like a ticking time bomb that no one wants to be holding when the time comes for it to explode.
Like West Coast, Collingwood's game is built on the foundations of those who came before – including past Collingwood teams – but with a genuine smallball forward set and a deeper midfield than those who've come before.
Both clubs can switch it up if required. As we discussed on Monday, the Eagles midfield has dialled up two extraordinary efforts on the deck in its past two games; Collingwood kicked the ball as much as it has all year against Richmond on Friday night.
The best feature of this chess match is both clubs will think the best way to win is to double down on their respective strengths; the strengths that exist as a natural counter to each other's game. Fortunately we've seen this play out already, in the second qualifying final three weeks ago.
West Coast were in front early, Collingwood hit back and manage to wrest control of the game, and it took an uncharacteristically stoic effort from West Coast's midfield to get them over the top.
Collingwood's tactical shift took Richmond by surprise last weekend. Will they look to disrupt West Coast's plans by being a little more careful with the ball? They certainly weren't in the qualifying final, committing a turnover roughly every four disposals (24 per cent, versus 18.4 per cent during the home and away season).
Indeed, Collingwood's fifth worst and third worst turnover rates have been recorded against West Coast in 2018. The Eagles are only the 12th-best turnover-creating team in the league when adjusting for opposition disposals. This could prove a fatal issue for the Pies if not addressed ahead of the game, for two reasons.
First is the simple fact West Coast counterattacks with quick kicks to forwards held ahead of the play. This gives the Eagles the means to swiftly punish mistakes; word is the groundskeepers are still cleaning up the disembowelled guts of Melbourne from last weekend.
Compounding this is the second consideration: Collingwood's commitment to the ball in attack leaves their back line exposed in the event of quick turnovers.
Overwhelming the opposition in attack is critical to Collingwood's ability to score, but the Eagles are well placed to convert that strength into a weakness should Collingwood fail to take care of the ball.
That's not to say the Pies shouldn't do what they do best. However, if West Coast hits them with the sort of frontal midfield force it has displayed in their past five quarters of football and the turnovers begin, Collingwood might need to turn to a bit of ball control to help settle down.
Either that or Collingwood turns to one of its clear personnel advantages: Brodie Grundy. Grundy was one of the best handful of players in 2018 (don't let his Brownlow votes fool you), and looms as large over this game as any of the 44 players to suit up.
That's because he's excellent, but also because the ruck is a bit of a sore spot for West Coast. Scott Lycett and Nathan Vardy. Last time out, the two Eagles combined for 21 disposals, 28 hitouts, four clearances and one goal. Grundy had 18 disposals, 48 hitouts, five clearances and no goals on his own.
Lycett's ruck work has fallen away precipitously in the back end of the year, the free agent winning more than half of his ruck contests just once since he missed Round 16's game against GWS.
Last time out against the Pies Lycett was involved in 42 contests, winning 14 hit outs. Vardy had been performing better albeit on fewer ruck contests, but has won just 36 per cent of his ruck contests in the past three weeks.
In their two prior matchups, Grundy was involved in 65 ruck contests (Round 17) and 77 ruck contests (qualifying final) respectively. His season average is 71 contests.
Getting closer to that season average, or going through it, would afford the Pies an ability to gain some extra territory from stoppages. And that may help lean against the turnover challenge West Coast will throw down.
Without Brad Sheppard and with Will Schofield, West Coast's defensive set is on the taller side of neutral. That looms as a potential source of advantage for the Pies.
Sheppard went down in the qualifying final, and it meant Chris Masten was sent back to play a more defensive role in West Coast's backline (rather than the flanker role he's performed this season).
He didn't do much that showed up on the stats sheet, but he was essentially there to help lock up some space and help direct Collingwood's ball movement away from the dangerous spot inside 50.
West Coast has no natural match up for both Jaidyn Stephenson or Jordan De Gooey. Tom Cole gets one of them at all times, but the other will find themselves on Shannon Hurn, Chris Masten or another one of West Coast's mid-sized players. That's a problem for West Coast, because they are all slow on the deck.
Collingwood's plan will almost certainly be to try and create one on one mismatches with these two deep inside 50, and attempt to use their respective speed and acceleration to run West Coast's defenders out of the game.
The Pies won't kick long and high inside 50, because the Eagles own the air down that end of the ground – Mason Cox backing up from his other-worldly game against Richmond notwithstanding.
While we're on the topic of surprising forwards, Chris Mayne's second half of the year has been superb, and there's no natural matchup for him in West Coast's defensive half given the need to contain so many threatening players.
Mayne laid a team-leading eight tackles in the qualifying final, including three inside Collingwood's attacking 50. He only needs that plus a couple of goals to be an influential player.
There's been some talk that Brody Mihocek will ostensibly act as a tagger on West Coast's interceptor-in-chief Jeremy McGovern.
That's a smart play, particularly if the Pies manage to make him accountable and create even a little bit of doubt in McGovern's mind.
However I would expect coach Adam Simpson will instruct McGovern to play his natural game and make Collingwood force the issue.
Mihocek kicked 2.3 last time out, and the same in Round 17. It's an interesting little subplot in an area of the ground that looms large.
You've seen the stat: West Coast is a stellar 12-0 (with a percentage of 151.5 per cent) when both Jack Darling and Josh Kennedy play. When just one of them played, the Eagles are 6-6 (with a percentage of 102.7 per cent).
They duo aren't the only factor behind that record – it also has to do with location of games, opponent and form – but the difference is pretty stark. The lads over at HPN Footy covered this well at the start of the finals series.
Tyson Goldsack held Kennedy well in the first half of the qualifying final. Though it has to be said Kennedy was also extraordinarily rusty in his first game back from a lower leg injury, and kicked 2.4 in the second half of the game.
He will certainly get the stopping task again, with aerial support from Jeremy Howe (who'll attempt to roll off the resting ruckman as his primary match up). Tom Langdon looms as the likely Darling matchup given his relative mobility.
The pair are so difficult to contain because they are both excellent leading marks – Darling only earning that title this year.
West Coast's forward line is adept at blocking and creating leading lanes for Kennedy in particular to work in, and if it isn't working inside the stripe then they're both happy to hunt further up the ground.
If they're both on point, it's hard to see West Coast being kept to a relatively low score.
In their respective preliminary finals, both West Coast and Collingwood battered their opponents with early pressure. That has been the Pies' stock in trade in 2018: they've won 15 of 25 first quarters, and gone on to win all but two of those games (whereas they've lost six of the ten games they were trailing at the first change).
Given the above, we know Collingwood will target the first five minutes of the game as their opportunity to disrupt West Coast's plans and get the game they want.
We saw that happen in the second quarter of the qualifying final, and it took the aforementioned effort from West Coast's midfield to pull them back into the game.
West Coast was sublime in its first quarter against Melbourne, but that was as much a product of Melbourne failing to show up. Indeed, just 18 tackles were laid in that first quarter, while the Eagles had 12 scoring shots.
Three weeks ago the game was in something of a stalemate early, until the Eagles kicked the first three goals of the game. That was ultimately the margin.
Grand finals tend to unfold like that. We have the early tension, before someone lands a decisive blow or two, and the game begins its ebbs and flows from there.
Collingwood will know if it can turn that early tension into a couple of goals, it will be on the front foot and have a chance to get into its game.
As implied by the column, this matchup has plenty of intrigue and tactical interplay. The above doesn't include the generic midfield-centric analysis, because it's fair to say whichever team can perform at its best through the middle will put itself into a sound position.
Collingwood would appear to have the wood on West Coast on paper, but the Eagles midfield group has been on a tear recently which we must respect. Andrew Gaff would have been an equaliser, but alas.
I see the midfield battle as a slight plus to Collingwood, but with a reasonable margin of error around it that it's hard to predict which way it'll go.
However, it'll all make for a compelling game, that I think will stay close for most of the afternoon. Ahead of the big boys getting involved, the market favours the Pies by 7.5 points, which is perhaps a reflection of the home ground advantage we discussed above and not much else.
My prediction? Here's a handful.
West Coast by 18 points.
Norm Smith Medal: Jack Redden.
Collingwood will lead at halftime.
Best of luck to all concerned. Except for Collingwood. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,818 |
So good it is. Pennsylvania Dutch road food. Zinn's has grown from a diner into a tourist "attraction" but fortunately still serves good Pennsylvania Dutch cooking.
Shown above - Chicken Noodle and Corn Soup and Turkey Croquettes with carrots and noodles. My favorite wasn't on the menu that day - Chicken and Waffles - waffles smothered in creamed chicken. Other dishes worth trying - Chicken Pot Pie, Pork and Sauerkraut and for the strong of stomach, Beef Heart and Pig Stomach. There's always a crock of homemade apple butter to spread on the rolls or bread.
For dessert - molasses based Shoo Fly pie. Try it once so you can say you've had it. Next time go any of the less cloyingly sweet home baked pies. I had rubarb my last time there.
That's Amos on the right - a giant Farmer that somewhat resembles founder Lee Zinn. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,873 |
Q: Why the limit is $\frac{x}{1+x}$ and not 1 I'm solving a problem about a plug flow reactor and I have this limit to compute. Just to control my result I asked Wolfram and I'm confuse can you explain me the result please.
I precise $x$ is a fixed value.
$$\lim_{R \to +\infty} \frac{1-\exp\left(\frac{x}{R+1}\right)}{\frac{R}{R+1}-\exp\left(\frac{x}{R+1}\right)}$$
When I made my reasonning I said, when $R$ goes to the infinity then the exponential terms both go to zero then the limit is the limit of $(1+R)/R$ which goes to $1$ when $R$ goes to the infinity.
So I bet my reasonning is false but don't know why. Please don't answer me with the L'Hopital theorem I dislike it.
Thank you in advance for your answer.
A: You cannot consider different expressions depending on the same variable one by one!
With your same reason the well known sequence $\left(1 + \frac{1}{n}\right)^n$ would converge to $1$ because the expression in the brackets tends to 1 and $1^n = 1$. But this is wrong and $$\lim_{n\to \infty} \left(1 + \frac{1}{n}\right)^n = e$$ holds.
And so you cannot consider each expression depending on $R$ by it's own.
Also you cannot use limit rules here which leads to an "$\frac{0}{0}$" result, and although you dislike L'Hopital theorem it should be the theorem of choice here…
A: Set $x/(R+1)=t$, so $R+1=x/t$ and
$$
\frac{R}{R+1}=\frac{x/t-1}{x/t}=\frac{x-t}{x}
$$
Then you have, depending on whether $x>0$ or $x<0$, the limit for $t\to0^+$ or the limit for $t\to0^-$. Let's compute the two-sided limit:
$$
\lim_{t\to0}\frac{1-e^t}{\frac{x-t}{x}-e^t}=
\lim_{t\to0}x\frac{1-e^t}{x-t-xe^t}=
\lim_{t\to0}x\frac{1-1-t+o(t)}{x-t-x-xt+o(t)}=\frac{x}{1+x}
$$
For $x=0$, the limit poses no problem and is $0$, so the formula is valid for $x\ne-1$.
For $x=-1$, we have
$$
\lim_{t\to0^{-}}\frac{1-e^t}{1+t-e^t}=-\infty
$$
A: One has
$$
\exp(h)=1+h+h\epsilon(h)
$$
with $\lim_{h\rightarrow 0} \epsilon(h)=0$ then, as $R\rightarrow +\infty$ one has
$$
\exp\left(\frac{x}{R+1}\right)=1+\frac{x}{R+1}+\frac{x}{R+1}\epsilon_1(R)
$$
with $\lim_{R\rightarrow +\infty} \epsilon_1(R)=0$. Substituting this (exact !) expression in your quotient gives the result.
A: Use the fundamental limit:
$$\lim_{x\rightarrow \infty}x(e^{1/x}-1)=1 \Rightarrow \lim_{x\rightarrow \infty}x(e^{a/x}-1)=a$$
and write your limit like:
$$\lim_{R \rightarrow \infty}\left(\frac{(R+1)(1-e^{x/(R+1)})}{-1+(R+1)(1-e^{x/(R+1)})}\right)=\frac{x}{1+x}$$
A: You should use L'Hopital theorem. That gives us:
$$\lim_{R\rightarrow+\infty} = \frac{\frac{d}{dR}\bigl( 1 - \exp(\frac{x}{R+1}) \bigr)}{\frac{d}{dR}\bigl( \frac{R}{R+1} - \exp(\frac{x}{R+1} \bigr)}$$
This is equal to:
$$\lim_{R\rightarrow+\infty} \frac{\frac{x}{(R+1)^2}}{ \frac{R+1-R}{(R+1)^2}+\frac{x}{(R+1)^2}} = \frac{x}{x+1}$$
I'm not 100% sure about this, but I think that you must use L'Hopital theorem if you have indeterminate form, like $\frac{0}{0}$ etc.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,318 |
\section*{Acknowledgement}}{}
\newenvironment{romenumerate}{\begin{enumerate
\renewcommand{\labelenumi}{\textup{(\roman{enumi})}}%
\renewcommand{\theenumi}{\textup{(\roman{enumi})}}%
}{\end{enumerate}}
\newcounter{oldenumi}
\newenvironment{romenumerateq
{\setcounter{oldenumi}{\value{enumi}}
\begin{romenumerate} \setcounter{enumi}{\value{oldenumi}}}
{\end{romenumerate}}
\newenvironment{thmxenumerate}{\begin{enumerate
\renewcommand{\labelenumi}{\textup{(\roman{enumi})}}%
\renewcommand{\theenumi}{\textup{(\roman{enumi})}}%
\setlength{\leftmargin}{0pt}
\setlength{\itemindent}{0pt}
}{\end{enumerate}}
\newcounter{thmenumerate}
\newenvironment{thmenumerate}
{\setcounter{thmenumerate}{0}%
\renewcommand{\thethmenumerate}{\textup{(\roman{thmenumerate})}}%
\def\item{\pa
\refstepcounter{thmenumerate}\textup{(\roman{thmenumerate})\enspace}}
}
{}
\newcounter{xenumerate}
\newenvironment{xenumerate}
{\begin{list}
{\upshape(\roman{xenumerate})}
{\setlength{\leftmargin}{0pt}
\setlength{\rightmargin}{0pt}
\setlength{\labelwidth}{0pt}
\setlength{\itemindent}{\labelsep}
\setlength{\topsep}{0pt}
\usecounter{xenumerate}} }
{\end{list}}
\newcommand\pfitem[1]{\par(#1):}
\newcommand\step[2]{\smallskip\noindent\emph{Step #1: #2} \noindent}
\newcommand{\refT}[1]{Theorem~\ref{#1}}
\newcommand{\refC}[1]{Corollary~\ref{#1}}
\newcommand{\refL}[1]{Lemma~\ref{#1}}
\newcommand{\refR}[1]{Remark~\ref{#1}}
\newcommand{\refS}[1]{Section~\ref{#1}}
\newcommand{\refP}[1]{Problem~\ref{#1}}
\newcommand{\refD}[1]{Definition~\ref{#1}}
\newcommand{\refE}[1]{Example~\ref{#1}}
\newcommand{\refF}[1]{Figure~\ref{#1}}
\newcommand{\refApp}[1]{Appendix~\ref{#1}}
\newcommand{\refTab}[1]{Table~\ref{#1}}
\newcommand{\refand}[2]{\ref{#1} and~\ref{#2}}
\newcommand\marginal[1]{\marginpar{\raggedright\parindent=0pt\tiny #1}}
\newcommand\SJ{\marginal{SJ}}
\newcommand\linebreakx{\unskip\marginal{$\backslash$linebreak}\linebreak}
\begingroup
\count255=\time
\divide\count255 by 60
\count1=\count255
\multiply\count255 by -60
\advance\count255 by \time
\ifnum \count255 < 10 \xdef\klockan{\the\count1.0\the\count255}
\else\xdef\klockan{\the\count1.\the\count255}\fi
\endgroup
\newcommand\nopf{\qed}
\newcommand\noqed{\renewcommand{\qed}{}}
\newcommand\qedtag{\tag*{\qedsymbol}}
\DeclareMathOperator*{\sumx}{\sum\nolimits^{*}}
\DeclareMathOperator*{\sumxx}{\sum\nolimits^{**}}
\DeclareMathOperator*{\sumy}{\sum\nolimits^{*}}
\DeclareMathOperator*{\sumyq}{\sum\nolimits^{*\!\!\!}}
\newcommand{\sum_{i=0}^\infty}{\sum_{i=0}^\infty}
\newcommand{\sum_{j=0}^\infty}{\sum_{j=0}^\infty}
\newcommand{\sum_{k=0}^\infty}{\sum_{k=0}^\infty}
\newcommand{\sum_{k=1}^\infty}{\sum_{k=1}^\infty}
\newcommand{\sum_{k=200}^\infty}{\sum_{k=200}^\infty}
\newcommand{\sum_{k=200}^\infty}{\sum_{k=200}^\infty}
\newcommand{\sum_{j=100}^\infty}{\sum_{j=100}^\infty}
\newcommand{\sum_{\ell=0}^\infty}{\sum_{\ell=0}^\infty}
\newcommand{\sum_{\ell=1}^\infty}{\sum_{\ell=1}^\infty}
\newcommand{\sum_{m=0}^\infty}{\sum_{m=0}^\infty}
\newcommand{\sum_{n=0}^\infty}{\sum_{n=0}^\infty}
\newcommand\set[1]{\ensuremath{\{#1\}}}
\newcommand\bigset[1]{\ensuremath{\bigl\{#1\bigr\}}}
\newcommand\Bigset[1]{\ensuremath{\Bigl\{#1\Bigr\}}}
\newcommand\biggset[1]{\ensuremath{\biggl\{#1\biggr\}}}
\newcommand\xpar[1]{(#1)}
\newcommand\bigpar[1]{\bigl(#1\bigr)}
\newcommand\Bigpar[1]{\Bigl(#1\Bigr)}
\newcommand\biggpar[1]{\biggl(#1\biggr)}
\newcommand\lrpar[1]{\left(#1\right)}
\newcommand\bigsqpar[1]{\bigl[#1\bigr]}
\newcommand\Bigsqpar[1]{\Bigl[#1\Bigr]}
\newcommand\biggsqpar[1]{\biggl[#1\biggr]}
\newcommand\lrsqpar[1]{\left[#1\right]}
\newcommand\xcpar[1]{\{#1\}}
\newcommand\bigcpar[1]{\bigl\{#1\bigr\}}
\newcommand\Bigcpar[1]{\Bigl\{#1\Bigr\}}
\newcommand\biggcpar[1]{\biggl\{#1\biggr\}}
\newcommand\lrcpar[1]{\left\{#1\right\}}
\newcommand\bigabs[1]{\bigl|#1\bigr|}
\newcommand\abs[1]{|#1|}
\newcommand\Bigabs[1]{\Bigl|#1\Bigr|}
\newcommand\biggabs[1]{\biggl|#1\biggr|}
\newcommand\lrabs[1]{\left|#1\right|}
\def\rompar(#1){\textup(#1\textup)}
\newcommand\xfrac[2]{#1/#2}
\newcommand\xpfrac[2]{(#1)/#2}
\newcommand\parfrac[2]{\lrpar{\frac{#1}{#2}}}
\newcommand\bigparfrac[2]{\bigpar{\frac{#1}{#2}}}
\newcommand\Bigparfrac[2]{\Bigpar{\frac{#1}{#2}}}
\newcommand\biggparfrac[2]{\biggpar{\frac{#1}{#2}}}
\newcommand\xparfrac[2]{\xpar{\xfrac{#1}{#2}}}
\newcommand\expX[1]{\exp\Bigl(#1\Bigr)}
\newcommand\expx[1]{\exp\bigl(#1\bigr)}
\newcommand\expQ[1]{e^{#1}}
\def\xexp(#1){e^{#1}}
\newcommand\ceil[1]{\lceil#1\rceil}
\newcommand\floor[1]{\lfloor#1\rfloor}
\newcommand\fract[1]{\{#1\}}
\newcommand\setn{\set{1,\dots,n}}
\newcommand\ntoo{\ensuremath{{n\to\infty}}}
\newcommand\Ntoo{\ensuremath{{N\to\infty}}}
\newcommand\asntoo{\text{as }\ntoo}
\newcommand\ktoo{\ensuremath{{k\to\infty}}}
\newcommand\stoo{\ensuremath{{s\to\infty}}}
\newcommand\bmin{\wedge}
\newcommand\norm[1]{\|#1\|}
\newcommand\bignorm[1]{\bigl\|#1\bigr\|}
\newcommand\Bignorm[1]{\Bigl\|#1\Bigr\|}
\newcommand\downto{\searrow}
\newcommand\upto{\nearrow}
\newcommand\half{\tfrac12}
\newcommand\thalf{\tfrac12}
\newcommand\iid{i.i.d.\spacefactor=1000}
\newcommand\ie{i.e.\spacefactor=1000}
\newcommand\eg{e.g.\spacefactor=1000}
\newcommand\viz{viz.\spacefactor=1000}
\newcommand\etc{etc.\spacefactor=1000}
\newcommand\cf{cf.\spacefactor=1000}
\newcommand{a.s.\spacefactor=1000}{a.s.\spacefactor=1000}
\newcommand{a.e.\spacefactor=1000}{a.e.\spacefactor=1000}
\renewcommand{\ae}{\vu}
\newcommand\whp{{w.h.p.\spacefactor=1000}}
\newcommand\ii{\mathrm{i}}
\newcommand{\longrightarrow}{\longrightarrow}
\newcommand\dto{\overset{\mathrm{d}}{\longrightarrow}}
\newcommand\pto{\overset{\mathrm{p}}{\longrightarrow}}
\newcommand\asto{\overset{\mathrm{a.s.}}{\longrightarrow}}
\newcommand\eqd{\overset{\mathrm{d}}{=}}
\newcommand\neqd{\overset{\mathrm{d}}{\neq}}
\newcommand\op{o_{\mathrm p}}
\newcommand\Op{O_{\mathrm p}}
\newcommand\bbR{\mathbb R}
\newcommand\bbC{\mathbb C}
\newcommand\bbN{\mathbb N}
\newcommand\bbT{\mathbb T}
\newcommand\bbZ{\mathbb Z}
\newcounter{CC}
\newcommand{\CC}{\stepcounter{CC}\CCx}
\newcommand{\CCx}{C_{\arabic{CC}}}
\newcommand{\CCdef}[1]{\xdef#1{\CCx}}
\newcounter{cc}
\newcommand{\cc}{\stepcounter{cc}\ccx}
\newcommand{\ccx}{c_{\arabic{cc}}}
\newcommand{\ccdef}[1]{\xdef#1{\ccx}}
\renewcommand\Re{\operatorname{Re}}
\renewcommand\Im{\operatorname{Im}}
\newcommand\E{\operatorname{\mathbb E{}}}
\renewcommand\P{\operatorname{\mathbb P{}}}
\newcommand\Var{\operatorname{Var}}
\newcommand\Cov{\operatorname{Cov}}
\newcommand\Corr{\operatorname{Corr}}
\newcommand\Exp{\operatorname{Exp}}
\newcommand\Po{\operatorname{Po}}
\newcommand\Bin{\operatorname{Bin}}
\newcommand\Be{\operatorname{Be}}
\newcommand\Ge{\operatorname{Ge}}
\newcommand\NBi{\operatorname{NegBin}}
\newcommand\Res{\operatorname{Res}}
\newcommand\fall[1]{^{\underline{#1}}}
\newcommand\rise[1]{^{\overline{#1}}}
\newcommand\argmax{\operatorname{argmax}}
\newcommand\ga{\alpha}
\newcommand\gb{\beta}
\newcommand\gd{\delta}
\newcommand\gD{\Delta}
\newcommand\gf{\varphi}
\newcommand\gam{\gamma}
\newcommand\gG{\Gamma}
\newcommand\gk{\varkappa}
\newcommand\gl{\lambda}
\newcommand\gL{\Lambda}
\newcommand\go{\omega}
\newcommand\gO{\Omega}
\newcommand\gs{\sigma}
\newcommand\gss{\sigma^2}
\newcommand\gth{\theta}
\newcommand\varepsilon{\varepsilon}
\newcommand\ep{\varepsilon}
\newcommand\bJ{\bar J}
\newcommand\cA{\mathcal A}
\newcommand\cB{\mathcal B}
\newcommand\cC{\mathcal C}
\newcommand\cD{\mathcal D}
\newcommand\cE{\mathcal E}
\newcommand\cF{\mathcal F}
\newcommand\cG{\mathcal G}
\newcommand\cH{\mathcal H}
\newcommand\cI{\mathcal I}
\newcommand\cJ{\mathcal J}
\newcommand\cK{\mathcal K}
\newcommand\cL{{\mathcal L}}
\newcommand\cM{\mathcal M}
\newcommand\cN{\mathcal N}
\newcommand\cO{\mathcal O}
\newcommand\cP{\mathcal P}
\newcommand\cQ{\mathcal Q}
\newcommand\cR{{\mathcal R}}
\newcommand\cS{{\mathcal S}}
\newcommand\cT{{\mathcal T}}
\newcommand\cU{{\mathcal U}}
\newcommand\cV{\mathcal V}
\newcommand\cW{\mathcal W}
\newcommand\cX{{\mathcal X}}
\newcommand\cY{{\mathcal Y}}
\newcommand\cZ{{\mathcal Z}}
\newcommand\tJ{{\tilde J}}
\newcommand\ett[1]{\boldsymbol1[#1]}
\newcommand\etta{\boldsymbol1}
\newcommand\smatrixx[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)}
\newcommand\limn{\lim_{n\to\infty}}
\newcommand\limN{\lim_{N\to\infty}}
\newcommand\qq{^{1/2}}
\newcommand\qqq{^{1/3}}
\newcommand\qqqq{^{1/4}}
\newcommand\qqw{^{-1/2}}
\newcommand\qqqw{^{-1/3}}
\newcommand\qqqqw{^{-1/4}}
\newcommand\qw{^{-1}}
\newcommand\qww{^{-2}}
\newcommand\qqqb{^{2/3}}
\newcommand\qqqbw{^{-2/3}}
\newcommand\qqc{^{3/2}}
\newcommand\qqcw{^{-3/2}}
\renewcommand{\=}{:=}
\newcommand\intoi{\int_0^1}
\newcommand\intot{\int_0^t}
\newcommand\intoo{\int_0^\infty}
\newcommand\intoooo{\int_{-\infty}^\infty}
\newcommand\intx[1]{\int_{#1-\infty\ii}^{#1+\infty\ii}}
\newcommand\intgs{\intx{\gs}}
\newcommand\oi{[0,1]}
\newcommand\ooo{[0,\infty)}
\newcommand\oooo{(-\infty,\infty)}
\newcommand\setoi{\set{0,1}}
\newcommand\dtv{d_{\mathrm{TV}}}
\newcommand\dd{\,\textup{d}}
\newcommand{probability generating function}{probability generating function}
\newcommand{moment generating function}{moment generating function}
\newcommand{characteristic function}{characteristic function}
\newcommand{uniformly integrable}{uniformly integrable}
\newcommand\rv{random variable}
\newcommand\lhs{left-hand side}
\newcommand\rhs{right-hand side}
\newcommand\Ai{\mathrm{Ai}}
\newcommand\AI{\mathrm{AI}}
\newcommand\Bi{\mathrm{Bi}}
\newcommand\Gi{\mathrm{Gi}}
\newcommand\Hi{\mathrm{Hi}}
\newcommand\AS[1]{\cite[#1]{AS}}
\newcommand\glx{a}
\newcommand\intakoo{\int_{a_k}^\infty}
\newcommand\eith{e^{\ii \gth}}
\newcommand\eigf{e^{\ii \gf}}
\newcommand\intg{\int_{\Gamma}}
\newcommand\intgg{\int_{\gGG}}
\newcommand\intgn{\int_{\Gamma_N}}
\newcommand\intggn{\int_{\gGG_N}}
\newcommand\zz{|z|}
\newcommand\vx[1]{V_{#1}}
\newcommand\vs{\vx{s}}
\newcommand\ws{W_{s}}
\newcommand\tM{\widetilde M}
\newcommand\xc{_{\gamc}}
\newcommand\ftm{f_{\tau,\tM}}
\newcommand\fm{f_{\tM}}
\newcommand\gc{g\xc}
\newcommand\cgc{\check g\xc}
\newcommand\kc{k\xc}
\newcommand\hcx[1]{h_{\gamc,\,#1}}
\newcommand\hcyc{\hcx{y+\gamc s^2}}
\newcommand\hca{\hcx{a}}
\newcommand\fcqqq{(4\gamc)\qqq}
\newcommand\fcqqqw{(4\gamc)\qqqw}
\newcommand\bccqqq{(2\gamc^2)\qqq}
\newcommand\bccqqqw{(2\gamc^2)\qqqw}
\newcommand\FT{\widehat}
\newcommand\gfc{c}
\newcommand\EN{S^{N}}
\newcommand\EM{S^{M}}
\newcommand\gGG{\gG'}
\newcommand\lcdots{\ldots\cdot}
\newcommand\gamc{\gamma}
\newcommand\Ex{\E_x}
\newcommand\tg{\widetilde g}
\newcommand\thx{\widetilde h}
\newcommand\tk{\widetilde k}
\newcommand\hthx{\widehat h}
\newcommand\htk{\widehat k}
\newcommand\ftt{f_\tau}
\newcommand\fttt{f_\tau(t)}
\newcommand\citetq[2]{\citeauthor{#2} \cite[{\frenchspacing #1}]{#2}}
\newcommand\gfo{\gf_0}
\newcommand\gfoo{\gf_\infty}
\newcommand\hatt{\widehat}
\newcommand\intiooioo{\int_{-\ii\infty}^{\ii\infty}}
\newcommand\gsb{b}
\newcommand\la{\lambda}
\newcommand{2^{1/3}}{2^{1/3}}
\newcommand{2^{2/3}}{2^{2/3}}
\newcommand{\sqrt{2}}{\sqrt{2}}
\newcommand{\sqrt{n}}{\sqrt{n}}
\newcommand{\sqrt{\pi}}{\sqrt{\pi}}
\newcommand{\varphi}{\varphi}
\newcommand{\infty}{\infty}
\newcommand{\left [}{\left [}
\newcommand{\right ]}{\right ]}
\newcommand{\left (}{\left (}
\newcommand{\right )}{\right )}
\newcommand{\lceil}{\lceil}
\newcommand{\rceil}{\rceil}
\newcommand{[\hspace{-0.5 mm}[}{[\hspace{-0.5 mm}[}
\newcommand{]\hspace{-0.5 mm}]}{]\hspace{-0.5 mm}]}
\newcommand{\nonumber}{\nonumber}
\newcommand{\begin{array}}{\begin{array}}
\newcommand{\end{array}}{\end{array}}
\newcommand{\begin{itemize}}{\begin{itemize}}
\newcommand{\end{itemize}}{\end{itemize}}
\newcommand{\begin{enumerate}}{\begin{enumerate}}
\newcommand{\end{enumerate}}{\end{enumerate}}
\newcommand{\begin{align}}{\begin{align}}
\newcommand{\begin{align*}}{\begin{align*}}
\newcommand{\begin{skip}}{\begin{skip}}
\newcommand{\end{align}}{\end{align}}
\newcommand{\end{align*}}{\end{align*}}
\newcommand{\end{skip}}{\end{skip}}
\newcommand{\quad}{\quad}
\newcommand{\mbox{$\mathbb V$}}{\mbox{$\mathbb V$}}
\newcommand{H\"older}{H\"older}
\newcommand\CS{Cauchy--Schwarz}
\newcommand\CSineq{\CS{} inequality}
\newcommand{L\'evy}{L\'evy}
\newcommand\ER{Erd\H os--R\'enyi}
\newcommand{Lov\'asz}{Lov\'asz}
\newcommand{\texttt{Maple}}{\texttt{Maple}}
\newcommand\ms[1]{\texttt{[ms #1]}}
\newcommand\XXX{XXX \marginal{XXX}}
\newcommand\REM[1]{{\raggedright\texttt{[#1]}\par\marginal{XXX}}}
\newenvironment{OLD}{\Small \REM{Old stuff to be edited:}\par}{}
\newenvironment{comment}{\setbox0=\vbox\bgroup}{\egroup}
\newcommand\citex{\REM}
\newcommand\refx[1]{\texttt{[#1]}}
\newcommand\xref[1]{\texttt{(#1)}}
\hyphenation{Upp-sala}
\newcommand\urladdrx[1]{{\urladdr{\def~{{\tiny$\sim$}}#1}}}
\begin{document}
\title[Moments of the location of the maximum]
{Moments of the location of the maximum of Brownian motion with parabolic drift}
\date{18 September 2012}
\author{Svante Janson}
\address{Department of Mathematics, Uppsala University, PO Box 480,
SE-751~06 Uppsala, Sweden}
\email{svante.janson@math.uu.se}
\urladdrx{http://www.math.uu.se/~svante/}
\thanks{Partly supported by the Knut and Alice Wallenberg Foundation}
\subjclass[2010]{60J65}
\begin{abstract}
We derive integral formulas, involving the Airy function, for moments of the
time a two-sided Brownian motion with parabolic drift attains its maximum.
\end{abstract}
\maketitle
\section{Introduction}\label{S:intro}
Let $W(t)$ be a two-sided Brownian motion with $W(0)=0$;
i.e., $(W(t))_{t\ge0}$ and
$(W(-t))_{t\ge0}$ are two independent standard Brownian
motions.
Fix $\gamma>0$, and consider the Brownian motion with parabolic drift
\begin{equation}\label{wg}
W_\gamma(t)\=W(t)-\gamma t^2.
\end{equation}
We are interested in
the maximum
\begin{equation}\label{mg}
M_\gamma\=\max_{-\infty<t<\infty} W_\gamma(t)
\end{equation}
of $W_\gam$ (which is a.s.\spacefactor=1000{} finite), and, in particular,
the \emph{location} of the maximum, which we denote by
\begin{equation}\label{zg}
V_\gam\=\argmax_t\bigpar{W_\gam(t)};
\end{equation}
in other words, $V_\gam=t \iff W_\gam(t)= M_\gam$.
(The maximum in \eqref{mg} is a.s.\spacefactor=1000{} attained at a unique point,
so $V_\gam$ is well-defined a.s.)
The parameter $\gam$ is just a scale parameter, see \refS{Scale}, so it can
be fixed arbitrarily without loss of generality.
The distribution of $V_\gam$ was called \emph{Chernoff's distribution} by
\citet{GW01} since it apparently first appeared in \citet{Chernoff}.
It has been studied by several authors;
in particular, \citet{G85,G89} gave a description of the
distribution
and \citet{GW01} give more explicit analytical and numerical formulas;
see also \citet{DS85}.
It has many applications in statistics,
see for example \citet{GW01} and the references given there, or, for a more
recent example, \citet{Anevski}.
The purpose of the present paper is to give formulas for the moments of
$V_\gam$ in terms of integrals involving the Airy function
$\Ai(x)$.
(Recall that $\Ai(x)$ satisfies
$\Ai''(x)=x\Ai(x)$
and is
up to a constant factor
the unique solution
that tends to 0 as $x\upto+\infty$.
See further \cite[10.4]{AS}.)
All odd moments of $V_\gam$ vanish by symmetry, and
our main result is the following formula for the even moments,
proved in \refS{Spf}.
(The special case $n=2$ is given by \citet{G11}.)
\begin{theorem}\label{T1}
For every even positive integer $n$, there is a polynomial
$p_n$ of degree at most $n/2$ such that
\begin{equation}\label{evk}
\E V_\gam^n
=\frac{2^{-n/3}\gam^{-2n/3}}{2\pi\ii}
\intiooioo \frac{p_n(z)}{\Ai(z)^2}{\dd z}
=\frac{2^{-n/3}\gam^{-2n/3}}{2\pi}
\intoooo \frac{p_n(\ii y)}{\Ai(\ii y)^2}{\dd y}.
\end{equation}
In particular, the variance of $V_\gam$ is
\begin{equation}\label{ev}
\E V_\gam^2=
-\frac{2^{-2/3}\gam^{-4/3}}{6\pi\ii} \intiooioo \frac{z}{\Ai(z)^2}{\dd z}
=\frac{2^{-2/3}\gam^{-4/3}}{6\pi\ii} \intoooo \frac{y}{\Ai(\ii y)^2}{\dd y}.
\end{equation}
\end{theorem}
The integrals are rapidly converging and easily computed numerically by
standard software.
The polynomials $p_n(z)$ can be found explicitly for any given $n$ by the
procedure in \refS{Spf}, but we do not know any general formula.
They are given for small $n$ in \refTab{Tab}.
See further the conjectures and problems in \refS{S?}.
Numerical values of the first ten absolute moments are given by \citet{GW01};
the first four were computed by Groeneboom and Sommeijer (1984, unpublished).
(Their values for the even moments agree with our formula.)
The maximum $M_\gam$ also appears in many applications.
Its distribution is given in \citet{G85,G89,G10}
and \citet{DS85},
see also,
for example,
\citet{B75,B81},
\citet{D74,D89},
\citet{SJ243}.
(\citet{G89} describes even the
joint distribution of the maximum $M_\gam$ and its location $V_\gam$, see also
\citet{DS85}.)
Formulas for the mean are given by \citet{DS85}, see also
\citet{SJ243}; in particular
\begin{equation}\label{em}
\E M_\gam=
-\frac{2^{-2/3}\gam^{-1/3}}{2\pi\ii} \intiooioo \frac{z}{\Ai(z)^2}{\dd z}.
\end{equation}
Comparing \eqref{em} and \eqref{ev}, we find the simple relation
\cite{G11}
\begin{equation}
\E V_\gam^2=\frac1{3\gam}\E M_\gam.
\end{equation}
Since $M_\gam=W_{V_\gam}-\gam V_\gam^2$, this implies that, at the maximum
point,
\begin{align}
\E W_{V_\gam}=\frac4{3}\E M_\gam
\qquad\text{and}\qquad
\E \gam V_\gam^2=\frac1{3}\E M_\gam
= \frac14 \E W_{V_\gam}.
\end{align}
It would be interesting to find a direct proof of these simple relations.
Formulas for second and higher moments of $M_\gam$ are given in
\citet{SJ243}; however, they are more complicated and do not correspond to
the formulas for moments of $V_\gam$ in the present paper.
It would be interesting to find relations between higher moments of $M_\gam$
and moments of $V_\gam$.
Let us finally mention that the random variable $V_\gam$ is the value at a
fixed time (0, to be precise)
of the stationary stochastic process
\begin{equation}
V_\gam(x)\=\argmax_t\bigpar{W(t)-\gam(t-x)^2},
\end{equation}
which is studied by
\citet{G85,G89,G11}. It would be interesting to find formulas for joint
moments of $V_{\gam}(x)$, in particular for the covariances.
\section{Scaling}\label{Scale}
For any $a>0$, $W(at)\eqd a\qq W(t)$ (as processes on
$(-\infty,\infty)$), and thus
\begin{align}
V_\gam&= a \argmax \bigpar{W(at)-\gamma (at)^2}
\notag\\&
\eqd a\argmax \bigpar{a\qq W(t)-a^2\gamma t^2}
=aV_{a\qqc\gam}. \label{b1m}
\end{align}
The parameter $\gam$ is thus just a scale parameter, and it suffices
to consider a single choice of $\gam$.
Although the choices $\gam=1$ and $\gam=1/2$ seem most natural, we will use
$\gam=1/\sqrt2$ which gives simpler formulas. We thus define
$V\=V_{1/\sqrt2}$ and have, for any $\gam$,
\begin{equation}\label{vgam}
V_\gam\eqd2\qqqw\gam\qqqbw V.
\end{equation}
Similarly,
\begin{equation}
M_\gam\eqd \gam_1\qqq\gam\qqqw M_{\gam_1}
\end{equation}
for any $\gam,\gam_1>0$.
\section{First formulas for moments}
By \citet[Corollary 3.3]{G89}, $V$ has the density
\begin{equation}\label{vf}
f(x)=\tfrac12g(x)g(-x),
\end{equation}
where $g$ has the Fourier transform, see \cite[(3.8)]{G89}
(this is where our choice $\gam=2\qqw$ is convenient),
\begin{equation}\label{hg}
\hatt g(t)\=\intoooo e^{\ii tx}g(x)\dd x = \frac{2^{1/2}}{\Ai(\ii t)},
\qquad -\infty <t<\infty .
\end{equation}
Note that $|\Ai(\ii t)|\to\infty$ rapidly as $t\to\pm\infty$,
see \cite[10.4.59]{AS} or \cite[(A.3)]{SJ243},
so $\hatt g(t)$
is rapidly decreasing.
In fact, it follows from the precise asymptotic formula
\cite[10.4.59]{AS} that
\begin{equation}\label{sw}
\Ai(x+\ii y)\qw = O\bigpar{e^{-cy\qqc}}
\end{equation}
for some $c>0$, uniformly for $|x|\le A$ for any fixed $A$ and $|y|\ge1$,
say (to avoid the zeros of $\Ai$). By differentiation (Cauchy's estimates,
see \eg{} \cite[Theorem 10.25]{Rudin-real}),
it follows that the same holds for all derivatives
of $\Ai(x+\ii y)\qw$, and thus all derivatives of $\hatt g(t)$
decrease rapidly.
In particular, $\hatt g$
belongs to the Schwartz class $\cS$ of rapidly decreasing functions on
$\bbR$; since this class is preserved by the Fourier transform, also
$g\in\cS$
(see \eg{} \cite[Theorem 7.7]{Rudin-FA}).
In particular, $g$ is integrable.
The characteristic function of $V$ is the Fourier transform $\hatt f(t)$,
and thus by \eqref{vf},
\eqref{hg}
and
standard Fourier analysis
(see \eg{} \cite[Theorems 7.7--7.8]{Rudin-FA}, but note the different
normalization chosen there),
with $\check g(t)\=g(-t)$,
\begin{equation}\label{chf}
\begin{split}
\gf(t)
&=\hatt f(t)
=\frac12\widehat{g\check g}(t)
=\frac12\cdot\frac1{2\pi}\bigpar{\hatt g*\hatt{\check g}}
=\frac1{4\pi}{\hatt g*\check{\hatt g}}
\\
&=\frac1{2\pi}\intoooo\frac{\dd s}{\Ai(\ii(t+s))\Ai(\ii s)}
=\frac1{2\pi\ii}\intiooioo\frac{\dd z}{\Ai(z+\ii t)\Ai(z)}.
\end{split}
\end{equation}
This is also given in \citet[Lemma 2.1]{G11} (although with typos in the
formula).
The last integral is taken along the imaginary axis, but we can change the
path of integration, as long as it passes to the right of the zeros $a_n$ of
the Airy function (which are real and negative), for example a line $\Re
z=\gsb$ with
$\gsb>a_1=-|a_1|$.
We pause to note the following.
\begin{theorem}
\label{Tmgf}
The moment generating function $\E e^{tV}$
is an entire function of $t$, and is given by, for any
complex $t$,
\begin{equation}\label{mgf}
\E e^{tV}
=\frac1{2\pi\ii}\int_{\gsb-\ii\infty}^{\gsb+\ii\infty}
\frac{\dd z}{\Ai(z+ t)\Ai(z)},
\end{equation}
for any real $\gsb$ with $\gsb>a_1$ and $\gsb+\Re t>a_1$.
\end{theorem}
\begin{proof}
The density function $f(t) =\exp(-t^3/3)$ as
$t\to\pm\infty$ by \cite[Corollary 3.4(iii)]{G89}.
Hence, $\E e^{tV}<\infty$ for
every real $t$, which implies that $\E e^{tV}$ is an entire function of $t$.
The formula \eqref{mgf} now follows from \eqref{chf}
by analytic continuation (for each fixed $\gsb$).
\end{proof}
By differentiation of \eqref{chf} (or \eqref{mgf}) we obtain, for any $n\ge 0$,
\begin{equation}\label{vak}
\E V^n = (-\ii)^n \frac{\dd^n}{\dd t^n}\gf(0)
=\frac{1}{2\pi\ii}\intiooioo \frac{\dd^n}{\dd z^n}\parfrac{1}{\Ai(z)}
\frac{\dd z}{\Ai(z)}.
\end{equation}
By integration by parts, this is generalized to:
\begin{theorem}
For any $j,k\ge0$,
\begin{equation}\label{jut}
\E V^{j+k}
=\frac{(-1)^{j} }{2\pi\ii}
\intiooioo \frac{\dd^j}{\dd z^j}\parfrac{1}{\Ai(z)}
\cdot\frac{\dd^k}{\dd z^k}\parfrac{1}{\Ai(z)} {\dd z}.
\end{equation}
\end{theorem}
\begin{proof}
For $j=0$, this is \eqref{vak}.
If we denote the integral on the \rhs{} of \eqref{jut} by $J(j,k)$, then, for
$j,k\ge0$, integration by parts yields
$A(j,k)=-A(j-1,k+1)$, and the result follows by induction.
\end{proof}
Since $\E V^{j+k}=\E V^{k+j}$,
we see again by symmetry that $\E V^n=0$ when $n$ is odd.
For even $n$, it is natural to take $j=k=n/2$ in \eqref{jut}.
For small $n$, this yields the following
examples.
First, $n=j=k=0$ yields
\begin{equation}\label{ev0}
1= \E V^{0}
=\frac{1 }{2\pi\ii}
\intiooioo \frac{1}{\Ai(z)^2},
\end{equation}
as noted by \citet{DS85};
this is easily verified directly, since $\pi\Bi(z)/\Ai(z)$ is a
primitive function of $1/\Ai^2$, see \cite{AlbrightG86}.
Next, for $n=2$ and $n=4$ we get
\begin{equation}\label{ev2}
\begin{split}
\E V^{2}
=\frac{-1}{2\pi\ii}
\intiooioo \lrpar{\frac{\dd}{\dd z}\parfrac{1}{\Ai(z)}}^2
{\dd z}
=\frac{-1}{2\pi\ii} \intiooioo \frac{\Ai'(z)^2}{\Ai(z)^4}{\dd z}
\end{split}
\end{equation}
and
\begin{equation}\label{ev4}
\begin{split}
\E V^{4}
&=\frac{1}{2\pi\ii}
\intiooioo\lrpar{ \frac{\dd^2}{\dd z^2}\parfrac{1}{\Ai(z)}}^2
{\dd z}
\\&
=\frac{1}{2\pi\ii}
\intiooioo\lrpar{-\frac{z}{\Ai(z)}+ \frac{2\Ai'(z)^2}{\Ai(z)^3}}^2
{\dd z}.
\end{split}
\end{equation}
\begin{remark}\label{Rdiff}
Since $\Ai''(z)=z\Ai(z)$, it follows by induction that the $m$:th derivative
$\frac{\dd^m}{\dd z^m}\bigpar{\Ai(z)\qw}$ can be expressed as a linear
combination (with integer coefficients)
of terms
\begin{equation}\label{rdiff}
\frac{z^j\Ai'(z)^k}{\Ai(z)^\ell}
\end{equation}
with $j,k\ge0$, $2j+k\le m$ and $\ell=k+1$.
\end{remark}
\section{Some combinatorics of Airy integrals}\label{Scomb}
Inspired by \refR{Rdiff}, we define in general, for any integers
$j,k,\ell\ge0$ with $\ell>k$,
\begin{equation}
I(j,k,\ell)\=
\frac{1}{2\pi\ii} \intiooioo \frac{z^j\Ai'(z)^k}{\Ai(z)^\ell}{\dd z}.
\end{equation}
(The integrand decreases rapidly as $z\to\pm\ii\infty$ because $\ell>k$,
by \eqref{sw} and $\Ai'(z)/\Ai(z)\sim -z\qq$ as $|z|\to\infty$
in any sector $|\arg z|\le\pi-\varepsilon$ with $\varepsilon>0$
\cite[10.4.59 and 10.4.61]{AS};
thus the
integral is absolutely convergent.)
Then, recalling $\Ai''(z)=z\Ai(z)$, for any $j,k\ge0$ and $\ell>k$,
\begin{equation}
\begin{split}
0&=
\frac{1}{2\pi\ii} \intiooioo
\frac{\dd}{\dd z}\frac{z^j\Ai'(z)^k}{\Ai(z)^\ell}{\dd z}
\\&
=j I(j-1,k,\ell)+k I(j+1,k-1,\ell-1)-\ell I(j,k+1,\ell+1),
\end{split}
\end{equation}
where we for convenience define $I(j,k,\ell)=0$ for $j<0$ or $k<0$.
Consequently, for $j,k\ge0$ and $\ell>k$,
\begin{equation}
I(j,k+1,\ell+1)=\frac j\ell I(j-1,k,\ell)+\frac k\ell I(j+1,k-1,\ell-1),
\end{equation}
or, for $j\ge0$, and $\ell>k\ge1$,
\begin{equation}\label{e44}
I(j,k,\ell)=
\begin{cases}
\frac{j}{\ell-1} I(j-1,0,\ell-1),
& k=1,
\\
\frac{j}{\ell-1} I(j-1,k-1,\ell-1)
+\frac{k-1}{\ell-1} I(j+1,k-2,\ell-2),
& k\ge2.
\end{cases}
\end{equation}
By repeatedly using this relation, any $I(j,k,\ell)$ may be expressed as a
rational combination of $I(p,0,\ell-k)$ with $0\le p\le j+k/2$.
For example,
\begin{align}
I(0,1,3)&=0; \label{i013}\\
I(0,2,4)&=\tfrac13I(1,0,2);\label{i024}\\
I(1,2,4)&=\tfrac13I(0,1,3)+\tfrac13I(2,0,2)=\tfrac13I(2,0,2);\label{i124}\\
I(0,4,6)&=\tfrac35I(1,2,4)=\tfrac15I(2,0,2).\label{i046}
\end{align}
\section{Back to moments}\label{Spf}
\begin{proof}[Proof of \refT{T1}]
By combining
\eqref{vak} and \refR{Rdiff},
we can express any moment $\E V^n$ as a linear
combination with integer coefficients of terms
$I(j,k,k+2)$ with $2j+k\le n$.
By repeated use of \eqref{e44} (see the comment after it),
this can be further developed into
a linear combination with rational coefficients
of terms
$I(j,0,2)$ with $0\le j\le n/2$.
For example, by \eqref{ev2} and \eqref{i024},
\begin{equation}\label{ev2a}
\E V^2=-I(0,2,4)=-\frac13I(1,0,2)=
-\frac{1}{6\pi\ii} \intiooioo \frac{z}{\Ai(z)^2}{\dd z},
\end{equation}
which yields \eqref{ev} by \eqref{vgam}.
To continue with higher moments we have next, by \eqref{ev4} and
\eqref{i124}--\eqref{i046},
\begin{equation}\label{ev4a}
\begin{split}
\E V^4&=I(2,0,2)-4I(1,2,4)+4I(0,4,6)
\\&
=\Bigpar{1-\frac43+\frac45}I(2,0,2)=\frac{7}{15}I(2,0,2)
\\&
=\frac{7}{30\pi\ii} \intiooioo \frac{z^2}{\Ai(z)^2}{\dd z}.
\end{split}
\end{equation}
Similarly (using Maple),
\begin{equation}\label{ev6a}
\E V^6
=\frac{1}{2\pi\ii} \intiooioo \frac{(26-31z^3)/21}{\Ai(z)^2}{\dd z}.
\end{equation}
In general this procedure yields, for some rational numbers $b_{nj}$,
\begin{equation}\label{eva}
\E V^n
=\sum_{j=0}^{n/2} b_{nj}I(j,0,2)
=\frac{1}{2\pi\ii} \intiooioo \frac{p_n(z)}{\Ai(z)^2}{\dd z}.
\end{equation}
where $p_n(z)\=\sum_{j=0}^{n/2}b_{nj}z^j$ is a polynomial of degree at most $n/2$.
By \eqref{vgam}, this is equivalent to the more general \eqref{evk}.
\end{proof}
For odd $n$, we already know that $\E V^n=0$, so we are mainly interested in
$p_n$ for even $n$.
The polynomials $p_0,p_2,p_4,p_6$ are implicit in
\eqref{ev0}, \eqref{ev2a}, \eqref{ev4a} and \eqref{ev6a}, and some further
cases (computed with Maple) are given in \refTab{Tab}.
\begin{table}
\begin{align*}
p_0(z)&=1\\
p_2(z)&=-\tfrac13 z\\
p_4(z)&=\tfrac7{15}z^2\\
p_6(z)&=-\tfrac{31}{21}z^3+\tfrac{26}{21}\\
p_8(z)&=\tfrac{127}{15}z^4-\tfrac{196}9z\\
p_{10}(z)&=-\tfrac{2555}{33}z^5+\tfrac{13160}{33}z^2\\
p_{12}(z)&=
{\tfrac {1414477}{1365}}\,{z}^{6}
-{\tfrac {2419532}{273}}\,{z}^{3}+
{\tfrac {1989472}{1365}}.
\end{align*}
\caption{The polynomials $p_n(z)$ for small even $n$.}
\label{Tab}
\end{table}
\begin{remark}
We can see from \refTab{Tab} that
(for these $n$) $p_n(z)$ contains only terms $z^j$ where $j\equiv n/2\pmod3$.
This is easily verified for all even $n$:
a closer look at the induction in \refR{Rdiff} shows that only terms
\eqref{rdiff} with $2j+k\equiv m\pmod3$ appears,
and the reduction in \eqref{e44} preserves $2j+k\pmod 3$.
\end{remark}
\section{Problems and conjectures}\label{S?}
The proof above yields an algorithm for computing the polynomials $p_n(z)$,
but no simple formula for them. We thus ask the following.
\begin{problem}
Is there an explicit formula for the coefficients $b_{nj}$, and thus for
the polynomials $p_n(z)$? Perhaps a recursion formula?
\end{problem}
We have computed $p_n(z)$ for $1\le n\le100$ by Maple, and based on the results
(see also \refTab{Tab}),
we make the following conjectures.
\begin{conjectures}
\begin{romenumerate}
\item \label{codd}
$p_n(z)=0$ for every odd $n$.
\item \label{cexact}
$p_n(z)$ has degree exactly $n/2$; \ie, the coefficient $b_{n,n/2}$ of
$z^{n/2}$ is non-zero.
\item \label{csinh}
These leading coefficients have exponential generating function
\begin{equation}\label{gf}
\sum_{n=0}^\infty \frac{b_{n,n/2}}{ n!}x^n
=
\frac{x}{\sinh x}.
\end{equation}
\end{romenumerate}
\end{conjectures}
Of these conjectures, \ref{codd} is natural, since we know that $\E V^n=0$
for odd $n$, and \ref{cexact} is not surprising.
The precise conjecture \eqref{gf} is perhaps more surprising. We
have verified that the coefficients up to $x^{100}$ agree, but we have no
general proof.
The simple form of \eqref{gf} suggests also the following open problem.
\begin{problem}
Is there an explicit formula for the generating function
$\sum_{n=0}^\infty p_n(z)x^n$?
\end{problem}
\newcommand\AAP{\emph{Adv. Appl. Probab.} }
\newcommand\JAP{\emph{J. Appl. Probab.} }
\newcommand\JAMS{\emph{J. \AMS} }
\newcommand\MAMS{\emph{Memoirs \AMS} }
\newcommand\PAMS{\emph{Proc. \AMS} }
\newcommand\TAMS{\emph{Trans. \AMS} }
\newcommand\AnnMS{\emph{Ann. Math. Statist.} }
\newcommand\AnnPr{\emph{Ann. Probab.} }
\newcommand\CPC{\emph{Combin. Probab. Comput.} }
\newcommand\JMAA{\emph{J. Math. Anal. Appl.} }
\newcommand\RSA{\emph{Random Struct. Alg.} }
\newcommand\ZW{\emph{Z. Wahrsch. Verw. Gebiete} }
\newcommand\DMTCS{\jour{Discr. Math. Theor. Comput. Sci.} }
\newcommand\AMS{Amer. Math. Soc.}
\newcommand\Springer{Springer-Verlag}
\newcommand\Wiley{Wiley}
\newcommand\vol{\textbf}
\newcommand\jour{\emph}
\newcommand\book{\emph}
\newcommand\inbook{\emph}
\def\no#1#2,{\unskip#2, no. #1,}
\newcommand\toappear{\unskip, to appear}
\newcommand\webcite[1]
\texttt{\def~{{\tiny$\sim$}}#1}\hfill\hfill}
\newcommand\webcitesvante{\webcite{http://www.math.uu.se/~svante/papers/}}
\newcommand\arxiv[1]{\webcite{arXiv:#1.}}
\def\nobibitem#1\par{}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,204 |
\section{Introduction}
\setcounter{equation}{0}
In astrophysics a galaxy or a globular cluster is often modeled as a large ensemble
of particles, i.e., stars, which interact only by the gravitational field which they
create collectively, collisions among the stars being sufficiently rare to
be neglected. The time evolution of the distribution function $f=f(t,x,v)\geq 0$
of the stars in phase space is then given by the Vlasov-Poisson system:
\begin{equation}
\partial_{t}f+v\cdot \nabla_{x}f-\nabla_{x}U\cdot \nabla_{v}f =0,
\label{vlasov}
\end{equation}
\begin{equation}
\Delta U = 4\pi \rho,\ \lim_{|x| \to \infty} U(t,x) = 0, \label{poisson}
\end{equation}
\begin{equation}
\rho(t,x) = \int f(t,x,v)dv \label{rhodef}.
\end{equation}
Here $t\in\R$ denotes time, and $x,v\in \R^3$ denote position and velocity,
$U=U(t,x)$ is the gravitational potential of the ensemble, and $\rho=\rho(t,x)$
is its spatial density.
A well-known approach to obtaining steady states of the Vlasov-Poisson system
is to make an ansatz of the form
\[
f_{0}(x,v) = \phi(E),
\]
where the particle energy $E$ is defined as
\[
E:=\frac{1}{2}|v|^2+U_{0}(x).
\]
Since for a time-independent potential $U_0=U_{0}(x)$ the particle energy
is conserved along the characteristics of the Vlasov equation (\ref{vlasov})
it remains to make sure that for the chosen $\phi$ the resulting semilinear
Poisson equation
\begin{equation} \label{semilinpoisson}
\Delta U_0 = 4\pi \int \phi \left(\frac{1}{2}|v|^2+U_{0}\right)\, dv
\end{equation}
has a solution. For a very large class of ansatz functions $\phi$
this approach leads to steady states with finite mass and compact support,
cf.\ \cite{RR}. Steady states where as above
the distribution of the particles in phase space
depends only on the particle energy are usually called
{\em isotropic} and are always spherically symmetric.
A central questions in stellar dynamics, which has attracted considerable attention in the
astrophysics literature, cf.\ \cite{BT,FP} and the references there, is the
dynamical stability of such steady states.
The Vlasov-Poisson system is conservative, i.e., the total energy
\[
{\cal H} (f) := E_\mathrm{kin} (f) + E_\mathrm{pot} (f)
= \frac{1}{2} \int |v|^2 f(x,v)\,dv\,dx - \frac{1}{8 \pi} \int |\nabla U_f(x)|^2 dx
\]
of a state $f$ is conserved along solutions and hence is a natural candidate for a
Lyapunov function in a stability analysis; $U_f$ denotes the potential
induced by $f$. However, the energy does not have critical
points, i.e., the linear part in an expansion about
any state $f_0$ with potential $U_0$ does not vanish:
\[
{\cal H}(f) = {\cal H}(f_0) +
\int\!\!\!\!\int \left(\frac{1}{2} |v|^2 + U_0 \right)(f-f_0) \,dv\,dx
- \frac{1}{8 \pi} \int|\nabla U_f-\nabla U_0 |^2 dx .
\]
To remedy this situation one observes that
for any reasonable function $\Phi$ the so-called {\em Casimir functional}
\[
{\cal C}(f) := \int\!\!\!\!\int \Phi(f(x,v))\,dv\,dx
\]
is conserved as well. If the energy-Casimir functional
\[
{{\cal H}_C} := {\cal H} + {\cal C}
\]
is expanded about an isotropic steady state, then
\begin{eqnarray*}
{{\cal H}_C} (f)
&=&
{{\cal H}_C} (f_0) +
\int\!\!\!\!\int (E + \Phi'(f_0))\,(f-f_0) \,dv\,dx \nonumber \\
&&
{}- \frac{1}{8 \pi} \int|\nabla U_f-\nabla U_0 |^2 dx
+ \frac{1}{2} \int\!\!\!\!\int \Phi''(f_0) (f-f_0)^2 \,dv\,dx
+ \ldots \\
&=:&
{{\cal H}_C} (f_0) + D {{\cal H}_C}(f_0)[f-f_0] + D^2 {{\cal H}_C}(f_0)[f-f_0] + \ldots .
\end{eqnarray*}
Now one can try to choose $\Phi$ is such a way that at least formally
$f_0$ is a critical point of the energy-Casimir functional,
i.e., $\Phi^{\prime }(f_{0})= -E$. If $\phi' < 0$ on the support
of $f_0$ the former relation holds on this support if $\Phi' = -\phi^{-1}$.
The formal second order variation in the above expansion then takes the form
\begin{equation} \label{h2}
D^2{{\cal H}_C} (f_0)[g] = \frac{1}{2}\int\!\!\!\!\int_{\{f_{0}>0\}}
\frac{1}{-\phi'(E)} g^2\, dv\, dx
-\frac{1}{8\pi}\int |\nabla_{x}U_{g}|^2 dx.
\end{equation}
It is natural to expect that positive definiteness of this quadratic
form should
imply stability for $f_{0}$. Ever since the seminal work of {\sc Antonov} \cite{An}
there have been vigorous efforts in the astrophysics community to establish
this positive definiteness and to derive stability results in this way.
An important step in this direction was to show that the above
quadratic form is positive definite on linearized, dynamically accessible
perturbations. To make this precise
we define the Lie-Poisson bracket of two functions $f_1, f_2$
of $x$ and $v$ as
\begin{equation}
\{f_{1},f_{2}\} := \nabla _{x}f_{1}\cdot \nabla _{v}f_{2}-\nabla
_{v}f_{1}\cdot \nabla _{x}f_{2}. \label{lie}
\end{equation}
Then the following holds:
\begin{lemma} \label{ks}
Let $\phi^{\prime }<0$ and let $h\in
C_{c}^{\infty }(\R^6)$ be spherically symmetric with support in the set
$\{f_{0}>0\}$ and such that $h(x,-v)=-h(x,v)$. Then
\[
D^2{{\cal H}_C} (f_0)[\{f_{0},h\}]
\geq
-\frac{1}{2} \int_{f_{0}>0} \phi^{\prime }(E)
\left[ |x\cdot v|^2
\left\vert \left\{ E,\frac{h}{x\cdot v}\right\} \right\vert^2
+ \frac{1}{r}U_{0}^{\prime}\, h^2
\right] \,dv\,dx.
\]
\end{lemma}
Here $U_0'$ denotes the radial derivative of the steady state potential.
Since $U_0$ is radially increasing, the right hand side in the estimate above
is indeed positive for $h\neq 0$. We refer to \cite{KS,SDLP} for astrophysical
investigations where this result is used to analyze
linearized stability. We do not go into the reasons why perturbations of
the form $\{f_0,h\}$ are called dynamically accessible for the linearized system,
but for the sake of completeness we provide a proof of this elegant result in
the Appendix. Despite its significance the result is still quite a distance
away from a true, nonlinear stability result.
There are at least two serious mathematical
difficulties. Firstly, it is very challenging to use the positivity of
$D^2 {{\cal H}_C} (f_{0})[g]$ to control the higher order remainder in the expansion
of the energy-Casimir functional \cite{Wa}. This is due to the
non-smooth nature of $f_{0}=\phi(E)$ in all important examples. Secondly,
even if one succeeds in controlling the higher order terms, the positivity
of $D^2 {{\cal H}_C} (f_{0})[g]$ in the lemma is only valid for certain
perturbation of the form $g=\{f_{0},h\}$. This class of perturbations is
invariant under solutions of the linearized Vlasov-Poisson system,
but it is not invariant under solutions of the nonlinear system.
To overcome these difficulties a variational approach was
initiated by {\sc Wolansky} \cite{Wo1} and then developed
systematically by {\sc Guo}
and {\sc Rein} \cite{G1,G2,GR1,GR2,GR3,GR4,R1,R2,RG}. Their
method entirely avoids the delicate
analysis of the second order term $D^2 {{\cal H}_C} (f_{0})$ in
(\ref{h2}), and it has led to the first rigorous nonlinear stability
proofs for a large class of steady states. More precisely, a large class
of steady states is obtained as minimizers of energy-Casimir functionals
under a mass constraint $\int f = M$,
and their minimizing property then entails their stability.
In particular, all polytropes $f_{0}(x,v)=(E_{0}-E)_{+}^{k}$
with $0<k\leq 7/2$ are covered; here $E_0<0$ is a certain cut-off
energy, and $(\cdot)_+$ denotes the positive part.
For $k >7/2$ the corresponding steady state has infinite mass
and is therefore unphysical. In addition, many new stable galaxy models were
established. The variational method has also been investigated in
\cite{DSS,H,LMR,SS,Wo2}.
Despite its considerable success, the variational approach has drawbacks
and limitations, the main one being that by its very nature it can not
access the stability of steady states which are only local, but not global
minimizers of the energy-Casimir functional.
Since the existence of the steady state as a (global) minimizer
is aimed for, certain growth conditions on the Casimir function $\Phi$ are needed,
which are not satisfied for all steady states with $\phi' <0$.
Most notably, the King model obtained by
\[
f_0(x,v) = (e^{E_0 - E}-1)_+
\]
is the single most important model which is currently out of the reach.
It describes isothermal galaxies and is widely used in astrophysics.
The corresponding Casimir function (\ref{q})
has very slow growth for $f \to \infty$, and as a result the
variational method fails.
The aim of the present paper is to develop a new approach
to nonlinear stability
results for steady states which need not be global minimizers of the
corresponding energy-Casimir functional by exploiting Lemma~\ref{ks}.
Although we are aiming for a general approach,
we focus here on the King model and as a first step
establish its nonlinear stability against
spherically symmetric, dynamically accessible perturbations.
The paper proceeds as follows. In the next section we formulate our results.
The nonlinear stability of the King model is an easy corollary of
the following main theorem: In a certain neighborhood of the King model
the potential energy distance of a perturbation can be controlled
in terms of the energy-Casimir distance.
In particular, within a certain class of perturbations, which
is invariant under solutions of the nonlinear Vlasov-Poisson system,
the King model is a local minimizer of the corresponding energy-Casimir
functional.
The resulting stability estimate is more explicit that the
ones obtained by the variational approach. The main part of the work
is then done in Section~3 where the local minimizing property
of the King model is established. In an appendix we give a proof of
Lemma~\ref{ks}.
\section{Main results}
\setcounter{equation}{0}
We start with a steady state $f_0$ with induced potential
$U_0$ and spatial density $\rho_0$, satisfying the relation
\begin{equation}
f_{0}(x,v)=\phi_0(E) := \left( e^{E_{0}-E}-1\right) _{+},\ E:=\frac{1}{2}|v|^2+U_{0}(x). \label{king}
\end{equation}
The cut-off energy $E_{0}<0$ is a given negative constant and
$(\cdot)_+$ denotes the positive part.
The corresponding Casimir function in the sense of the introduction is
\begin{equation}
\Phi_{0}(f) := (1+f)\ln (1+f)-f. \label{q}
\end{equation}
The existence of such King models,
i.e., of suitable solutions of the resulting semilinear Poisson equation
(\ref{semilinpoisson}), is established in \cite{RR}. Such a model has
compact support
\[
\supp f_0 = \{(v,v) \in \R^6 \mid E(x,v) \leq E_0\} =: \{E \leq E_0\},
\]
and it is spherically symmetric. A state $f$ is called
{\em spherically symmetric}
if for any rotation $A\in \mathrm{SO}(3)$,
\[
f(x,v) = f(A x,A v),\ x,v \in \R^3.
\]
It is well known that non-negative, smooth, and compactly supported
initial data $f(0) \in C^1_c(\R^6)$ launch unique global smooth solutions
$t\mapsto f(t)$
of the Vlasov-Poisson system \cite{Pf,LP,Sch}.
If the initial datum is spherically symmetric then this symmetry is preserved,
and the modulus of the particle angular momentum squared,
\[
L:= |x \times v|^2 = |x|^2 |v|^2 - (x\cdot v)^2 ,
\]
is conserved along characteristics of the Vlasov equation.
Hence for any smooth function $\Phi$ such that $\Phi(0,L)=0,\ L\geq 0$,
the functional
\[
\int\!\!\!\!\int \Phi(f,L)\,dv\,dx
\]
is conserved along spherically symmetric solutions of the
Vlasov-Poisson system;
unless explicitly stated otherwise integrals $\int$ always extend over $\R^3$.
We consider the following class of perturbations:
\begin{eqnarray*}
\mathcal{S}_{f_{0}} := \Bigl\{f\in L^1(\R^6)
&\mid&
f\geq 0 \ \mbox{spherically symmetric}, \\
&& \int\!\!\!\!\int\Phi(f,L)=\int\!\!\!\!\int \Phi(f_{0},L)
\text{ for all }\Phi\in C^2([0,\infty[^2) \text{ with } \\
&&
\Phi(0,L)=\partial_f \Phi(0,L)= 0,\ L\geq 0,\ \mbox{and}\
\partial_f^2 \Phi \ \mbox{bounded}\Bigr\}.
\end{eqnarray*}
As noted above, the class $\mathcal{S}_{f_{0}} \cap C^1_c(\R^6)$ is
invariant under solutions of the Vlasov-Poisson system.
Moreover, functions in $\mathcal{S}_{f_{0}}$ are equi-measurable to
$f_0$, i.e., for every $\tau>0$ the sets $\{ f>\tau\}$ and $\{ f_0>\tau\}$ have
the same measure,
in particular, $||f||_p = ||f_0||_p$ for any $L^p$-norm, $p\in [1,\infty]$.
For the Casimir function $\Phi_0$ defined in (\ref{q})
we define the energy-Casimir functional
as in the introduction.
Then for $f\in \mathcal{S}_{f_{0}}$ we have
\begin{eqnarray*}
{{\cal H}_C}(f)-{{\cal H}_C}(f_{0})
&=&
\int\!\!\!\!\int [\Phi_{0}(f)-\Phi_{0}(f_{0})+(E-E_{0})(f-f_{0})]\,dv\,dx \\
&&-\frac{1}{8\pi }\int |\nabla U_{f}-\nabla U_0|^2 dx;
\end{eqnarray*}
notice that $\int f = \int f_0$ which allows us to bring
in the term $E_0(f-f_0)$. Now
\[
(E-E_0) (f-f_0) \geq -\Phi_0'(f_0) (f-f_0)
\]
with equality on the support of $f_0$, and hence for
$f\in \mathcal{S}_{f_{0}}$,
\begin{eqnarray} \label{convest}
\Phi_{0}(f)-\Phi_{0}(f_{0})+(E-E_{0})(f-f_{0})
&\geq&
\frac{1}{2} \inf_{0\leq \tau \leq ||f_0||_\infty} \Phi_0''(\tau)\; (f-f_0)^2 \nonumber \\
&\geq&
C_0 (f-f_0)^2
\end{eqnarray}
where $C_0:= 1/(2+ 2 ||f_0||_\infty)$; notice again that
$||f||_\infty = ||f_0||_\infty$ for any
$f\in \mathcal{S}_{f_{0}}$.
The deviation from the steady state is going to be measured
by the quantity
\begin{eqnarray} \label{d}
d(f,f_{0})
&:=&
\int\!\!\!\!\int
[\Phi_{0}(f)-\Phi_{0}(f_{0})+(E-E_{0})(f-f_{0})]\,dv\,dx \nonumber \\
&&
{}+\frac{1}{8\pi }
\int |\nabla U_{f}-\nabla U_0|^2 dx,
\end{eqnarray}
which, as we have seen, controls both $||f-f_0||_2$ and
$||\nabla U_{f}-\nabla U_0||_2$,
and satisfies the following relation to the energy-Casimir functional:
\begin{equation} \label{decrel}
d(f,f_{0}) = {{\cal H}_C}(f)-{{\cal H}_C}(f_{0}) +
\frac{1}{4\pi }\int |\nabla U_{f}-\nabla U_0|^2 dx.
\end{equation}
Our stability result is the following:
\begin{theorem} \label{main}
There exist constants $\delta >0$ and $C>0$ such that
for any solution $t\mapsto f(t)$ of the Vlasov-Poisson system
with $f(0) \in \mathcal{S}_{f_{0}}\cap C_{c}^{1}(\R^6)$ and
\[
d(f(0),f_{0}) \leq \delta
\]
the estimate
\[
d(f(t),f_{0}) \leq C \; d(f(0),f_{0})
\]
holds for all time $t>0$.
\end{theorem}
\noindent
{\bf Remark:}
In order to better understand the perturbation class $\mathcal{S}_{f_{0}}$
we show that it contains
{\em spherically symmetric, dynamically accessible
perturbations}
by which we mean the following:
Let $H=H(x,v) \in C^2(\R^6)$ be spherically symmetric,
and let $g=g(s,x,v)$ denote the solution of the linear problem
\[
\partial_s g + \nabla_v H\cdot \nabla_x g - \nabla_x H\cdot \nabla_v g = 0,
\quad \mbox{i.e.},\quad
\partial_s g(s) = \{H,g(s)\},
\]
with initial datum $g(0) = f_0$; we assume that $H$ is such that
this solution exists on some interval $I$ about $s=0$. Then for any $s \in I$,
$g(s)$ is spherically symmetric and equi-measurable with $f_0$.
Moreover, for any function $\Phi$ as considered in the definition
of $\mathcal{S}_{f_{0}}$,
\[
\partial_s \Phi(g(s),L) = \partial_f \Phi(g(s),L) \{ H,g(s)\},
\]
and by a simple computation, $\{H,L\} = 0$.
Hence
\begin{eqnarray*}
\frac{d}{ds} \int\!\!\!\!\int \Phi(g(s),L)\,dv\,dx
&=&
\int\!\!\!\!\int
\left( \partial_f \Phi \{H,g(s)\} + \partial_L \Phi \{H,L\}\right)\,dv\,dx \\
&=&
\int\!\!\!\!\int \{H,\Phi(g(s),L)\}\,dv\,dx = 0
\end{eqnarray*}
after an integration by parts.
This means that $g(s) \in \mathcal{S}_{f_{0}}$ for any such
generating function $H$ and any $s$. The only undesirable restriction
in the class $\mathcal{S}_{f_{0}}$
or in the generating functions $H$ respectively is the spherical symmetry
which hopefully can be removed in the future.
\smallskip
The stability result Theorem~\ref{main} is easily deduced from the
following theorem:
\begin{theorem} \label{lower}
There exist constants $\delta _{0}>0,$ and $C_0>0$ such that
for all $f\in \mathcal{S}_{f_{0}}$ with $d(f,f_{0})\leq \delta _{0}$
the following estimate holds:
\[
{{\cal H}_C}(f)-{{\cal H}_C}(f_{0})\geq C_0 ||\nabla U_{f}-\nabla U_0||_2^2.
\]
\end{theorem}
Before going into the proof of this theorem, which will occupy the rest of this
paper, we conclude this section by deducing our stability result from it.
\noindent
{\bf Proof of Theorem~\ref{main}}.
Let $\delta:= \delta_0 (1+1/(4 \pi C_0))^{-1}$
with $\delta_0$ and $C_0$ from Theorem~\ref{lower}. Consider a solution
$t\mapsto f(t)$ of the Vlasov-Poisson system with
$f(0) \in \mathcal{S}_{f_{0}}\cap C_{c}^{1}(\R^6)$ and
\[
d(f(0),f_{0}) \leq \delta < \delta_0.
\]
Then by continuity we can choose some maximal $t^\ast \in ]0,\infty]$ such that
\[
d(f(t),f_{0}) < \delta_0,\ t\in [0,t^\ast[.
\]
Now $f(t) \in \mathcal{S}_{f_{0}}$ for all $t$, and hence
Theorem~\ref{lower}, the relation (\ref{decrel}) of $d$ to the energy-Casimir
functional, and the fact that the latter is a conserved quantity
yield the following chain of estimates for $t\in[0,t^\ast[$:
\begin{eqnarray*}
d(f(t),f_0)
&=&
{{\cal H}_C}(f(t)) - {{\cal H}_C}(f_0) + \frac{1}{4\pi} ||\nabla U_{f(t)}-\nabla U_0||_2^2\\
&\leq&
{{\cal H}_C}(f(t)) - {{\cal H}_C}(f_0) + \frac{1}{4 \pi C_0} \left({{\cal H}_C}(f(t)) - {{\cal H}_C}(f_0)\right)\\
&=&
\left(1+\frac{1}{4 \pi C_0}\right) \, \left({{\cal H}_C}(f(0)) - {{\cal H}_C}(f_0)\right)\\
&\leq&
\left(1+\frac{1}{4 \pi C_0}\right)\,d(f(0),f_0) < \delta_0.
\end{eqnarray*}
This implies that $t^\ast = \infty$, and Theorem~\ref{main}
is established.
\prfe
\section{Proof of Theorem~\ref{lower}}
\setcounter{equation}{0}
Theorem~\ref{lower} is proven by contradiction.
There are two main ingredients: The first part (Subsection~\ref{redsec}) is a
general argument to establish that if the estimate in the theorem fails, then
there exists a non-zero function $g$ such that $D^2 {{\cal H}_C} (f_{0})[g]\leq 0$. The
second (Subsection~\ref{hsec}) is to use the measure-preserving property
incorporated in our perturbation class $\mathcal{S}_{f_{0}}$ to conclude that
$g=\{f_{0},h\}$ for some function $h$. This leads to a
contradiction to Lemma~\ref{ks} (Subsection~\ref{contrasec}).
\subsection{Existence of $g\neq 0$ with $D^2 {{\cal H}_C} (f_{0})[g]\leq 0$}
\label{redsec}
The aim of this subsection is to prove the following result:
\begin{lemma} \label{reduction}
Assume that Theorem \ref{lower} were false. Then there is a
function $g\in L^2(\R^6)$ which is spherically symmetric,
supported in the set $\{E \leq E_0\}$, even in $v$, i.e., $g(x,-v)=g(x,v)$, and
such that
\begin{equation} \label{normal}
\frac{1}{8\pi }||\nabla U_{g}||_2^2 =1,
\end{equation}
\begin{equation} \label{hle}
D^2 {{\cal H}_C} (f_{0})[g] = \frac{1}{2}\int\!\!\!\!\int_{\{f_{0}>0\}}
\Phi_{0}^{\prime \prime }(f_{0})\,g^2\,dv\,dx - 1 \leq 0,
\end{equation}
\begin{equation} \label{preserve}
\int\!\!\!\!\int \partial_f\Phi(f_{0},L)\,g\,dv\,dx = 0
\end{equation}
for all functions $\Phi$ as specified in the definition of
$\mathcal{S}_{f_{0}}$.
\end{lemma}
\noindent{\bf Proof.}\
If Theorem \ref{lower} were false, then for any $n\in \mathbb N$ there exists
$f_{n}\in \mathcal{S}_{f_{0}}$ such that
\[
d(f_{n},f_{0})<\frac{1}{n}
\]
but
\begin{equation} \label{hsmall}
{{\cal H}_C}(f_{n})-{{\cal H}_C}(f_{0}) < \frac{1}{8\pi\, n}
||\nabla U_{f_{n}}-\nabla U_0||_2^2,
\end{equation}
in particular, $f_n \neq f_0$.
We define
\begin{equation} \label{sigma}
\sigma_{n}:=\frac{1}{\sqrt{8\pi}} ||\nabla U_{f_{n}}-\nabla U_0||_2 ,\
g_{n} := \frac{1}{\sigma _{n}} \left(f_{n}-f_{0}\right)
\end{equation}
so that
\begin{equation} \label{normaln}
\frac{1}{8\pi }||\nabla U_{g_{n}}||_2^2 =1
\end{equation}
and $f_n = f_0 + \sigma_n g_n$.
By (\ref{decrel}) and (\ref{sigma}),
\begin{equation}
\sigma _{n}^2\leq d(f_{n},f_{0}) < \frac{1}{n}. \label{sigmazero}
\end{equation}
\noindent
{\em Bounds on $(g_n)$ and weak limit.}\\
As a first step in the
proof of the lemma we need to establish bounds on the
sequence $(g_n)$ which allow us extract a subsequence that
converges to some $g$ which will be our candidate for the
function asserted in the lemma.
By (\ref{sigma}), (\ref{d}), (\ref{decrel}), and (\ref{hsmall}) we find that
\begin{eqnarray} \label{q''}
&&
\frac{1}{\sigma _{n}^2}\int\!\!\!\!\int
[\Phi_{0}(f_{0}+\sigma_{n}g_{n})-\Phi_{0}(f_{0})+(E-E_{0})\sigma _{n}g_{n}]
\,dv\,dx - 1 \nonumber \\
&&
\qquad\qquad = \frac{1}{\sigma _{n}^2}\left(d(f_n,f_0)
-\frac{1}{4\pi } \int |\nabla U_{f_{n}}-\nabla U_0|^2 dx \right) \nonumber\\
&&
\qquad\qquad = \frac{1}{\sigma _{n}^2} \left({{\cal H}_C}(f_{n})-{{\cal H}_C}(f_{0})\right)
< \frac{1}{\sigma _{n}^2}
\frac{1}{8\pi\, n } ||\nabla U_{f_{n}}-\nabla U_0||_2^2
= \frac{1}{n}.\quad
\end{eqnarray}
Recalling the estimate (\ref{convest}) which is applicable since
$f_n \in \mathcal{S}_{f_{0}}$ this implies that
\[
1+\frac{1}{n} > \frac{1}{\sigma _{n}^2}\int\!\!\!\!\int [\ldots]\,dv\,dx
\geq C_0 \frac{1}{\sigma _{n}^2} \int\!\!\!\!\int |f_n-f_0|^2\,dv\,dx
= C_0\, \int\!\!\!\!\int |g_n|^2\,dv\,dx,
\]
which means that the sequence $(g_n)$ is bounded in $L^2(\R^6)$.
Moreover, since the integrand $[\ldots]$ in (\ref{q''}) is non-negative
we find that
\[
1+\frac{1}{n} > \frac{1}{\sigma _{n}^2}
\int\!\!\!\!\int_{\{E\geq E_0\}} [\ldots]\,dv\,dx
\geq \frac{1}{\sigma _{n}}
\int\!\!\!\!\int_{\{E\geq E_0\}} (E-E_0)\, g_n\,dv\,dx,
\]
where we used the fact that on the set $\{E\geq E_0\}$ the steady
state distribution $f_0$ and hence also $\Phi_0(f_0)$ vanish
while $\Phi_0(f_n)\geq 0$.
By (\ref{sigmazero}),
\begin{equation} \label{zerooutside1}
\int\!\!\!\!\int_{\{E\geq E_0\}} (E-E_0)\, g_n\,dv\,dx \leq 2 \sigma_n \to 0,\
n\to \infty.
\end{equation}
Now fix any $E_0< E_1 < 0$. Since $\lim_{|x|\to \infty} U_0 (x) = 0$
it follows that $E=E(x,v) > E_1$ for $x$ or $v$ large so that the set
$\{E\leq E_1\}\subset \R^6$
is compact. Hence $(g_n)$ is bounded in $L^1(\{E\leq E_1\})$.
In addition
\begin{equation} \label{zerooutside2}
\int\!\!\!\!\int_{\{E\geq E_1\}} g_n\,dv\,dx
\leq
\int\!\!\!\!\int_{\{E\geq E_0\}} \frac{E-E_0}{E_1-E_0}\, g_n\,dv\,dx \to 0;
\end{equation}
notice that $g_n \geq 0$ outside the support of $f_0$, i.e.,
on the set $\{E > E_0\}$.
Thus we have shown that $(g_n)$ is bounded in $L^1\cap L^2(\R^6)$.
We extract a subsequence, denoted again by $(g_n)$ such that
\[
g_n \rightharpoonup g\ \mbox{weakly in}\ L^2(\R^6).
\]
Since $g_n\geq 0$ on $\{E>E_0\}$ and since (\ref{zerooutside2}) holds
for any $E_0< E_1 < 0$ we conclude that $g$ vanishes a.~e.\ outside
the set $\{E\leq E_0\}$ as desired. Since the functions $g_n$
are spherically symmetric so is $g$.
\smallskip
\noindent
{\em Proof of (\ref{normal}).}\\
In order to pass the week convergence into (\ref{normaln})
we need better bounds for the sequence $(g_n)$.
Indeed, we can bound its kinetic energy:
\[
\int\!\!\!\!\int_{\{E\geq E_0\}}
\left(\frac{1}{2} |v|^2 +U_0(x) -E_0\right)\,g_n\,dv\,dx
= \int\!\!\!\!\int_{\{E\geq E_0\}}
\left(E-E_0\right)\,g_n\,dv\,dx \to 0
\]
by (\ref{zerooutside1}) so that
\[
\int\!\!\!\!\int_{\{E\geq E_0\}}
\frac{1}{2} |v|^2\,g_n\,dv\,dx
\]
is bounded, while
\[
\int\!\!\!\!\int_{\{E\leq E_0\}} \frac{1}{2} |v|^2\,g_n\,dv\,dx
\leq (E_0 - U_0(0)) \int\!\!\!\!\int_{\{E\leq E_0\}}
g_n\,dv\,dx
\]
is bounded as well; recall that $U_0$ is spherically symmetric
and radially increasing. Now well known interpolation arguments
imply that the sequence of induced spatial densities $(\rho_{g_n})$
is bounded in $L^1\cap L^{7/5} (\R^3)$,
cf.\ \cite[Ch.~1, Lemma~5.1]{R3}, so without loss of generality,
\[
\rho_{g_n} \rightharpoonup \rho_g\ \mbox{weakly in}\ L^{7/5}(\R^3).
\]
Let again $E_0<E_1<0$ be arbitrary but fixed and choose $R_1>0$
such that $U_0(R_1)=E_1$ which implies that $E(x,v)\geq E_1$
for $|x|\geq R_1$. Then by (\ref{zerooutside2}),
\[
\int_{\{|x|\geq R_1\}} |\rho_{g_n}|\, dx
\leq
\int\!\!\!\!\int_{\{E\geq E_1\}} g_n\,dv\,dx
\to 0.
\]
The fact that the sequence $(\rho_{g_n})$ remains concentrated in this
way gives the desired compactness:
\[
\nabla U_{g_n} \to \nabla U_g \ \mbox{strongly in}\ L^2(\R^3),
\]
cf.\ \cite[Ch.~2, Lemma~3.2]{R3}.
Hence we can pass to the limit in (\ref{normaln}) and find that
$g$ satisfies the condition (\ref{normal}) in the lemma.
\smallskip
\noindent
{\em Proof of (\ref{hle}).}\\
Since $||g_{n}||_2$ is bounded it follows from (\ref{sigmazero}) that
\[
||\sigma _{n}g_{n}||_2 \leq C\sigma _{n} \to 0.
\]
Therefore, after extracting again a subsequence,
$\sigma _{n}g_{n} \to 0$ almost
everywhere.
By Egorov's Theorem there exists for every
$m\in \mathbb N$ a measurable
subset $K_{m} \subset \{E \leq E_0\}$ with the property that
\[
\vol \left(\{E \leq E_0\}\setminus K_m\right) <\frac{1}{m}
\ \mbox{and}\
\lim_{n \to \infty}\sigma_{n}g_{n}=0 \
\mbox{uniformly on}\ K_{m};
\]
note that the set $\{E \leq E_0\}$ has finite measure. In addition we can
assume that $K_m \subset K_{m+1},\ m\in \mathbb N$.
On the set $\{E \leq E_0\}$,
\[
[\Phi_0(f_n) - \Phi_0(f_0) + (E-E_0) \sigma_n g_n]
= \frac{1}{2} \Phi_0''(f_0)(\sigma_n g_n)^2
+ \frac{1}{6} \Phi_0'''(f_0+\tau \sigma_n g_n)(\sigma_n g_n)^3
\]
for some $\tau \in [0,1]$. Since both $f_0$ and
$f_0+\sigma_n g_n = f_n$ are non-negative the same is true
for $f_0+\tau \sigma_n g_n$, and we can use the estimate
\[
|\Phi_0''' (f)| = \frac{1}{(1+f)^2} \leq 1,\ f\geq 0
\]
the estimate (\ref{q''}), and the fact that the integrand
$[\ldots]$ in (\ref{q''}) is
non-negative by (\ref{convest}) to conclude that
\begin{eqnarray*}
\int\!\!\!\!\int_{K_m} \frac{1}{2} \Phi_0''(f_0)\, |g_n|^2\,dv\,dx
&=&
\frac{1}{\sigma_n^2} \int\!\!\!\!\int_{K_m} [\ldots]\,dv\,dx\\
&&
{}- \frac{1}{\sigma_n^2} \int\!\!\!\!\int_{K_m}
\frac{1}{6} \Phi_0'''(f_0+\tau \sigma_n g_n)\,(\sigma_n g_n)^3\,dv\,dx\\
&<&
1+\frac{1}{n} + \sup_{K_m}|\sigma_n g_n| \int\!\!\!\!\int |g_n|^2\,dv\,dx.
\end{eqnarray*}
Taking the limit $n\to \infty $ in this estimate we find that
\[
\int\!\!\!\!\int_{K_m} \frac{1}{2} \Phi_0''(f_0)\, g^2\,dv\,dx \leq 1.
\]
If we observe the choice of the sets $K_m$, let $m\to \infty$,
and recall the fact that $g=0$ outside the set $\{E\leq E_0\}$
the proof of (\ref{hle}) is complete.
\smallskip
\noindent
{\em Proof of (\ref{preserve}).}\\
To prove (\ref{preserve}), the measure-preserving property of the set
$\mathcal{S}_{f_{0}}$ plays the crucial role.
Let $\Phi=\Phi(f,L)$ be a function as specified in
the definition of that set respectively in the lemma.
By Taylor expansion with respect to the first argument,
\[
\Phi(f_n,L)-\Phi(f_{0},L)
=\partial _{f}\Phi(f_{0},L)\,\sigma _{n}g_{n}+
\frac{1}{2}\partial_{f}^2\Phi(f_{0}+\tau\sigma_{n}g_{n},L)\,
(\sigma _{n}g_{n})^2
\]
for some $\tau\in[0,1]$. If we integrate this identity
and observe that, since $f_n \in \mathcal{S}_{f_{0}}$,
\[
\int\!\!\!\!\int \Phi(f_n,L)\,dv\,dx=\int\!\!\!\!\int \Phi(f_{0},L)\,dv\,dx,
\]
it follows that
\[
\int\!\!\!\!\int
\partial_{f}\Phi(f_{0},L)\,g_{n}\,dv\,dx
=
-\frac{1}{2}\sigma_{n}
\int\!\!\!\!\int\partial_f^2 \Phi(f_{0}+\tau \sigma_{n}g_{n},L)\,g_n^2\,dv\,dx
\to 0.
\]
As to the latter limit we note that $\partial_f^2 \Phi$ is bounded,
$(g_n)$ is bounded in $L^2(\R^6)$, and $\sigma _{n}\to 0$.
On the other hand $\partial_{f}\Phi(f_{0},L)$ is supported
on the compact set $\{E\leq E_0\}$ and hence bounded.
Since $g_{n}\to g$ in $L^2(\R^6)$, the identity (\ref{preserve})
follows as $n \to \infty$.
\smallskip
\noindent
{\em Conclusion of the proof of Lemma~\ref{reduction}.}\\
The function $g$ constructed above has all the properties
required in the lemma, except that it need not be even in $v$. Hence
we decompose it into its even and odd parts with respect to $v$: $g=g_\mathrm{even}+g_\mathrm{odd}$.
We claim that (\ref{normal}), (\ref{hle}), and (\ref{preserve})
remain valid for the even part. Since
\[
\rho_{g}=\int g\, dv=\int g_\mathrm{even}dv=\rho_{g_\mathrm{even}}
\]
we have $\nabla U_{g_\mathrm{even}}=\nabla U_{g}$, and (\ref{normal})
remains valid.
Since $\Phi_0^{\prime \prime }(f_0)$ is even in $v$,
\begin{eqnarray*}
1
&\geq&
\int\!\!\!\!\int \frac{1}{2}
\Phi_0''(f_0)\, (g_\mathrm{even}+g_\mathrm{odd})^2\,dv\,dx
=
\int\!\!\!\!\int \frac{1}{2}
\Phi_0''(f_0)\, \left((g_\mathrm{even})^2+(g_\mathrm{odd})^2\right)\,dv\,dx \\
&\geq&
\int\!\!\!\!\int \frac{1}{2}
\Phi_0''(f_0)\, (g_\mathrm{even})^2\,dv\,dx,
\end{eqnarray*}
i.e., (\ref{hle}) remains valid. Finally, let $\Phi$ be as in
the definition of the set $\mathcal{S}_{f_{0}}$. Then
$\partial_f \Phi(f_0,L)$ is even in $v$ so that in (\ref{preserve})
the odd part of $g$ drops out, and the assertions of Lemma~\ref{reduction}
hold for $g_\mathrm{even}$.
\prfe
\subsection{Characteristics and $g=\{f_{0},h\}$}\label{hsec}
In this subsection we construct a spherically symmetric
function $h$ such that $g=\{f_{0},h\}$.
To this end we need to introduce variables which are adapted to the
spherical symmetry:
\[
r:=|x|,\ w:= \frac{x\cdot v}{r},\ L:=|x\times v|^2;
\]
$w$ is the radial velocity, and $L$ has already been used above.
Any spherically symmetric function of $x$ and $v$ such as
the desired $h$ can be written
in terms of these variables, so $h=h(r,w,L)$. Then
\[
\{f_{0},h\}
=
\phi_0'(E) \{E,h\}
= - \phi_0'(E)\,\left[w\, \partial_r +
\left(\frac{L}{r^3}-U_0'(r)\right)\,\partial_w \right]\, h,
\]
and the equation we wish to solve for $h$ reads
\begin{equation} \label{heqn}
\left[w\, \partial_r +
\left(\frac{L}{r^3}-U_0'(r)\right)\,\partial_w \right]\, h= - \frac{1}{\phi_0'(E)}\,g.
\end{equation}
In order to analyze its characteristic system
\[
\dot r = w,\ \dot w = \frac{L}{r^3}-U_0'(r)
\]
we define for fixed $L>0$ the effective potential
\[
\Psi_L(r):= U_0(r) + \frac{L}{2 r^2}
\]
and observe that in terms of the variables $r,w,L$ the
conserved particle energy takes the form
\[
E=E(x,v)=E(r,w,L) = \frac{1}{2}w^2 +\Psi_L(r);
\]
$L$ only plays the role of a parameter here since $\dot L = 0$.
We need to analyze the effective potential $\Psi_L$.
The boundary condition for $U_0$ at infinity implies that
$\lim_{r\to \infty} \Psi_L(r)=0$, and clearly,
$\lim_{r\to 0} \Psi_L(r)=\infty$. Moreover,
\[
U_0'(r)=\frac{4\pi }{r^2}\int_{0}^{r}\rho_{0}(s)\,s^2 ds =:
\frac{m_0(r)}{r^2}>0,\ r>0,
\]
since $U_0$ is increasing, thus $U_0(0) < E_0$, and by (\ref{king}),
$\rho_{0}(0)>0$. Now
\[
\Psi_L'(r) = 0 \ \Leftrightarrow\ \frac{m_0(r)}{r^2}- \frac{L}{r^3} =0
\ \Leftrightarrow\ m_0(r)-\frac{L}{r} = 0,
\]
and since the left hand side of the latter equation
is strictly increasing for $L>0$ with
$\lim_{r\to \infty}(m_0(r)-L/r) = M >0$ and
$\lim_{r\to 0}(m_0(r)-L/r) = -\infty$ there exists a unique $r_L >0$
such that
\[
\Psi_L'(r_L)=0,\ \Psi_L'(r)<0\ \mbox{for}\ r<r_L,
\ \Psi_L'(r)>0\ \mbox{for}\ r>r_L.
\]
Moreover, since
\[
\frac{d}{dr} \left(m_0(r)-\frac{L}{r}\right)
= 4 \pi r^2 \rho_0(r) + \frac{L}{r^2} >0
\]
the implicit function theorem implies that the mapping
$]0,\infty[ \ni L \mapsto r_L$ is continuously differentiable.
For $r=r_L$,
\begin{equation} \label{psi''>0}
\Psi''_L (r_L) = - 2\frac{m_0(r)}{r^3} + 3\frac{L}{r^4}
+ 4 \pi \rho_0(r) = 4 \pi \rho_0(r) + \frac{L}{r^4} > 0.
\end{equation}
The behavior of $\Psi_L$ implies that for any $L>0$ and
$\Psi_L(r_L)<E<0$ there exist two unique
radii $0<r_{-}(E,L)<r_L<r_{+}(E,L)< \infty$ such that
\[
\Psi_L(r_\pm(E,L))=E,\ \mbox{and}\
\Psi_L(r)<E\ \Leftrightarrow\ r_{-}(E,L)<r<r_{+}(E,L).
\]
Since $\Psi'_L(r)\neq 0$ for $r \neq r_L$ it follows
again by the implicit function theorem that the mapping
$(E,L) \mapsto r_\pm (E,L)$ is $C^1$ on the set
$\{(E,L) \in \R \times ]0,\infty[ \mid \Psi_L(r_L) < E < 0 \}$.
Let $\tau\mapsto (r(\tau),w(\tau),L)$ be a characteristic curve with $L>0$ and
running in the set $\{E\leq E_0\}$, so that
$\Psi_L(r_L)<E\leq E_0 < 0$. Then
\[
r(\tau) \in [r_{-}(E,L),r_{+}(E,L)],\ w(\tau) = \pm \sqrt{2E - 2\Psi_L(r(\tau))}.
\]
By the equation (\ref{heqn}) which we want to solve,
\[
\frac{d}{d\tau} h(r(\tau),w(\tau),L) = -\frac{1}{\phi_0'(E)} g(r(\tau),w(\tau),L),
\]
which we rewrite in terms of the parameter $r$ as
\[
\frac{d}{dr} h(r,w(r),L) = -\frac{1}{\phi_0'(E)\,w(r)} g(r,w(r),L),\
w(r) = \pm \sqrt{2E - 2\Psi_L(r)}.
\]
Hence for $(r,w,L) \in \{E\leq E_0\}$ with $L>0$
we define $h$ as follows:
We let $E:= \frac{1}{2} w^2 + \Psi_L(r)$ so that
$r_{-}(E,L)\leq r \leq r_{+}(E,L)$, and
\begin{equation}
h(r,w,L) := - \mathrm{sign}\, w \frac{1}{\phi_0'(E)}\int_{r_{-}(E,L)}^{r}
\frac{g(s,\sqrt{2E-2\Psi_L(s)},L)}{\sqrt{2E-2 \Psi_L(s)}} ds. \label{hg}
\end{equation}
Outside the set $\{E\leq E_0\}$ we let $h=0$.
We need to make sure that we can consistently set
$h(r,0,L)=0$, i.e., the integral in the above definition
must vanish for $r=r_+(E,L)$. To this end let
$\Phi$ be as specified in the definition of the class
$\mathcal{S}_{f_{0}}$. We want to apply the change of variables
$(x,v) \mapsto (r,w,L) \mapsto (r,E,L)$ to the integral in (\ref{preserve}).
Now
\begin{equation} \label{xvtorel}
dx\,dv = 8\pi^2 dr\,dw\,dL = 8\pi^2
\frac{dr\,dE\,dL}{\sqrt{2E-2\Psi_L(r)}},
\end{equation}
where we note that $g$ is even in $v$ and hence in $w$
so that we can restrict the integral to $w>0$. We obtain
the identity
\begin{eqnarray*}
0
&=&
\int\!\!\!\!\int \partial_f\Phi(f_{0},L)\,g\,dv\,dx
=
8\pi^2 \int_0^\infty \int_0^\infty \int_0^\infty
\partial_f\Phi(f_{0},L)\,g\,dr\,dw\,dL\\
&=&
8\pi^2
\int\!\!\!\!\int_M
\int_{r_{-}(E,L)}^{r_+(E,L)}
\frac{g(r,\sqrt{2E-2\Psi_L(r)},L)}{\sqrt{2E-2 \Psi_L(r)}} dr\,
\partial_f\Phi(\phi_0(E),L)\,dE\,dL,
\end{eqnarray*}
where $M:=\{(E,L)(x,v)|f_0(x,v)>0\}$.
The class of test functions $\partial_f\Phi(\phi_0(E),L)$ is sufficiently
large to conclude that for almost all $E$ and $L$,
\[
\int_{r_{-}(E,L)}^{r_{+}(E,L)}\frac{g(r,\sqrt{2E-2\Psi_L(r)},L)}
{\sqrt{2E-2\Psi_L(r)}} dr =0
\]
as desired.
\subsection{Contradiction to Lemma~\ref{ks}}
\label{contrasec}
As defined above, $h$ need not be smooth or even integrable,
so in order to derive a contradiction to Lemma~\ref{ks}
we need to regularize it. In order to do so it should first
be noted that $h$ as defined in (\ref{hg}) is measurable,
which follows by Fubini's Theorem and the fact that by the change
of variables formula the function
\[
(s,r,E,L) \mapsto \frac{g(s,\sqrt{2 E -2\Psi_L(s)},L)}
{\sqrt{2 E -2\Psi_L(s)}}
\mathbf{1}_{[\Psi_L(r_L),E_0]}(E)\mathbf{1}_{[r_-(E,L),r_+(E,L)]}(s)
\mathbf{1}_{[0,r]}(s)
\]
is integrable; $r\leq \max\{|x| \mid (x,v) \in \supp f_0\}$.
\smallskip
\noindent{\em The cut-off $h$.}\\
As a first step in regularizing $h$ we define for $m$ large the set
\[
\Omega_{m}:= \left\{(x,v)\in \R^6 \mid
E \leq E_{0}-\frac{1}{m},\ L\geq \frac{1}{m}\right\}.
\]
We want to approximate $h$ by $h\mathbf{1}_{\Omega _{m}}$.
In order to analyze this approximation the following lemma will be useful:
\begin{lemma} \label{line}
There exists a constant $C_{m}>0$ such that for
$L\geq \frac{1}{m}$ and $\Psi_L(r_L) < E \leq E_0$,
\[
\int_{r_{-}(E,L)}^{r_{+}(E,L)}\frac{dr}{\sqrt{2 E-2 \Psi_L(r)}} <C_{m}.
\]
\end{lemma}
\noindent{\bf Proof.}\
We first establish the following auxiliary estimate:
For all $m\in \mathbb N$ there exists a constant $\eta_{m}>0$ such that
for all $L\geq 1/m$, $\Psi_L(r_L)< E\leq E_0$,
and $r\in[r_-(E,L),r_+(E,L)]$,
\begin{equation} \label{claim}
\frac{|\Psi'_L(r)|}{\sqrt{\Psi_L(r)-\Psi_L(r_L)}}
\geq \eta _{m}.
\end{equation}
To see this let $m\in \mathbb N$ and $L,E,r$ be as specified.
Then
\[
E_0 \geq E \geq \Psi_L(r) = U_0(r) + \frac{L}{2 r^2}
\geq U_0(0) + \frac{1}{2 m r^2}
\]
and hence $r\geq (2m(E_0 - U_0(0))^{-1/2}$.
Let $R:=\max\{|x| \mid (x,v)\in \supp f_0\}$, i.e., $U_0(R)=E_0$.
Then
\[
L \leq 2 r^2 \left(E_0 - U_0(r)\right)
\leq 2 R^2 \left(E_0 - U_0(0)\right).
\]
Hence if we assume that (\ref{claim}) were false,
we can find a sequence $(r_{n},L_{n})\to (\bar{r},\bar{L})$
in the set
$[(2m(E_0 - U_0(0))^{-1/2},R]\times[1/m, 2 R^2 \left(E_0 - U_0(0)\right)]$
such that
\[
\lim_{n\to \infty }\frac{\Psi'_{L_n}(r_n)}
{\sqrt{\Psi_{L_{n}}(r_{n}))- \Psi_{L_{n}}(r_{L_{n}})}}=0.
\]
If $\bar{r}\neq r_{\bar{L}}$ it follows that
$\Psi_{\bar L} (\bar r) > \Psi_{\bar L} (r_{\bar L})$ and
$\Psi'_{\bar L}(\bar r) = 0$ which is a contradiction
to the uniqueness of the minimizer $r_{\bar L}$ of
$\Psi_{\bar L}$. So assume that $\bar{r} = r_{\bar{L}}$.
Now recall that $\Psi'_{L_n}(r_{L_n})=0$.
By Taylor expansion at $r=r_{L_n}$ we find intermediate values
$\theta_n$ and $\tau_n$ between $r_n$ and $r_{L_n}$ such that
\[
\frac{|\Psi'_{L_{n}}(r_{n})|}
{\sqrt{\Psi_{L_{n}}(r_{n}) - \Psi_{L_{n}}(r_{L_n})}}
=
\frac{|\Psi''_{L_{n}}(\theta_{n})\,(r_{n}-r_{L_{n}})|}
{\sqrt{\frac{1}{2} \Psi''_{L_{n}}(\tau_{n})\,(r_{n}-r_{L_{n}})^2}}
\to
\sqrt{2 |\Psi''_{\bar L}(r_{\bar L})|} \neq 0,\ n\to \infty,
\]
where we recall (\ref{psi''>0}).
This contradiction completes the proof of (\ref{claim}).
To complete the proof of the lemma we split the integral as
\[
\int_{r_{-}(E,L)}^{r_{+}(E,L)}\frac{dr}{\sqrt{2E-2\Psi_L(r)}}
=\int_{r_{-}(E,L)}^{r_L}\ldots + \int_{r_L}^{r_{+}(E,L)}\ldots
=: I_1 + I_2.
\]
In the first term, we make a change of variables
$u=\sqrt{\Psi_L(r)-\Psi_L(r_L)}$ so that
$\frac{du}{dr}=\frac{1}{2 u}\Psi_L'(r) <0$ on $[r_{-}(E,L),r_L[$.
By (\ref{claim}),
\begin{eqnarray*}
I_1
&=&
\int_{\sqrt{E-\Psi_L(r_L)}}^{0}
\frac{1}{\sqrt{2(E-\Psi_L(r_L)-u^2)}}\frac{dr}{du}du \\
&\leq&
\frac{\sqrt{2}}{\eta _{m}}
\int_{0}^{\sqrt{E-\Psi_L(r_L)}}
\frac{du}{\sqrt{E-\Psi_L(r_L)-u^2}}
=
\frac{\sqrt{2}}{\eta _{m}}
\int_{0}^{1}\frac{ds}{\sqrt{1-s^2}}<\infty
\end{eqnarray*}
by a further change of variables $u=\sqrt{E-\Psi_L(r_L)} s$.
The same type of estimate holds for the second part
$I_2$ of the integral under investigation, and the proof of the
lemma is complete.
\prfe
We now show that the cut-off function $h\mathbf{1}_{\Omega _{m}}$
is square integrable and solves the equation
$\{f_{0},h\mathbf{1}_{\Omega _{m}}\} =g\mathbf{1}_{\Omega _{m}}$
in the sense of distributions, more precisely:
\begin{lemma} \label{dis}
For any $m\in \mathbb N$ large,
$h\mathbf{1}_{\Omega _{m}}\in L^2(\R^6)$, and
for any spherically symmetric test function
$\psi=\psi(r,w,L) \in C^1([0,\infty[\times \R \times [0,\infty[)$,
\[
\int_{E\leq
E_{0}}\{f_{0},\psi \}h\mathbf{1}_{\Omega _{m}}=-
\int_{E\leq E_{0}}g\mathbf{1}_{\Omega _{m}}\psi.
\]
\end{lemma}
\noindent{\bf Proof.}\
We first prove that $h\mathbf{1}_{\Omega _{m}}\in L^2(\R^6)$.
Since the integrand is even in $v$ we can apply
the change of variables (\ref{xvtorel}):
\begin{eqnarray*}
&&
\int\!\!\!\!\int \mathbf{1}_{\Omega _{m}} h^2 dv\,dx\\
&&
\qquad
= 8 \pi^2 \int\!\!\!\!\int_{S_m}
\int_{r_{-}(E,L)}^{r_{+}(E,L)}h^2(r,\sqrt{2E-2\Psi_L(r)},L)
\frac{dr}{\sqrt{2E-2\Psi_L(r)}}\,dE\,dL,
\end{eqnarray*}
where
\begin{equation} \label{sm}
S_m:=
\left\{(E,L)=(E,L)(x,v)
\mid (x,v)\in \supp f_0,\ E\leq E_0-\frac{1}{m},\ L\geq \frac{1}{m}\right\}.
\end{equation}
Let $(E,L) \in S_m$. In the estimates below we write
$w(r)=\sqrt{2E-2\Psi_L(r)}$ and $r_\pm(E,L) = r_\pm$
for brevity. Then by the definition
(\ref{hg}) of $h$ and the fact that $1/|\phi_0'(E)| \leq 1$ for $E\leq E_0$,
\begin{eqnarray*}
\int_{r_{-}}^{r_{+}}h^2(r,w(r),L) \frac{dr}{w(r)}
&=&
\int_{r_{-}}^{r_{+}}
\left[\frac{1}{\phi_0'(E)} \int_{r_{-}}^{r}
g(s,w(s),L) \frac{ds}{w(s)}\right]^2
\frac{dr}{w(r)} \\
&\leq&
\int_{r_{-}}^{r_{+}}
\left[\int_{r_{-}}^{r_{+}}
g(s,w(s),L) \frac{ds}{w(s)}\right]^2
\frac{dr}{w(r)} \\
&\leq&
\left(\int_{r_{-}}^{r_{+}}\frac{dr}{w(r)}\right)^2
\int_{r_{-}}^{r_{+}}
g^2(r,w(r),L) \frac{dr}{w(r)}\\
&\leq&
C_m^2
\int_{r_{-}}^{r_{+}}
g^2(r,w(r),L) \frac{dr}{w(r)};
\end{eqnarray*}%
in the last two estimates
we used the Cauchy-Schwarz inequality and Lemma~\ref{line}.
A further integration with respect to $E$ and $L$ and the change
of variables $(r,E,L)\mapsto (x,v)$ shows that
$||h\mathbf{1}_{\Omega _{m}}||_2$ is bounded in terms of $C_m$ and
$||g||_2$.
Now let $\psi=\psi (r,w,L)$ be a test function
as specified in the lemma. Along characteristic curves
of (\ref{heqn}), which as before we parameterize by $r$
distinguishing between $w>0$ and $w<0$,
a simple computation shows that
\[
\{E,\psi\}= -\mathrm{sign}\, w \sqrt{2 E- 2 \Psi_L(r)}
\frac{d}{dr}[\psi (r,\mathrm{sign}\, w \sqrt{2 E- 2 \Psi_L(r)},L)].
\]
By the change of variables used repeatedly above, using the
abbreviation $w(r):=\sqrt{2 E- 2 \Psi_L(r)}$, and recalling
the definition (\ref{sm}) we find that
\begin{eqnarray*}
&&
\int\!\!\!\!\int \{f_{0},\psi \}h\mathbf{1}_{\Omega _{m}}dv\,dx
=\int_{\{w>0\}} \ldots + \int_{\{w<0\}} \ldots\\
&&
= -\int\!\!\!\!\int_{S_m} \phi_0'(E)
\int_{r_{-}}^{r_{+}} w(r)
\frac{d}{dr}[\psi (r,w(r),L)]\,
h(r,w(r),L) \frac{dr}{w(r)} dE\,dL\\
&& {}-\int\!\!\!\!\int_{S_m} \phi_0'(E)
\int_{r_{-}}^{r_{+}} (-w(r))
\frac{d}{dr}[\psi (r,-w(r),L)]\,
h(r,-w(r),L) \frac{dr}{w(r)} dE\,dL\\
&&
= \int\!\!\!\!\int_{S_m} \phi_0'(E)
\int_{r_{-}}^{r_{+}} \psi (r,w(r),L)\,
\left(-\frac{g(r,w(r),L)}{\phi_0'(E)\, w(r)}\right)\,dr\, dE\,dL\\
&&
{}- \int\!\!\!\!\int_{S_m} \phi_0'(E)
\int_{r_{-}}^{r_{+}} \psi (r,-w(r),L)\,
\frac{g(r,-w(r),L)}{\phi_0'(E)\, w(r)}dr\, dE\,dL\\
&&
= - \int\!\!\!\!\int \psi g \mathbf{1}_{\Omega _{m}}dv\,dx;
\end{eqnarray*}
notice that $h(r,\pm w(r),L)=0$ for $r=r_\pm(E,L)$, which
together with the definition (\ref{hg}) of $h$ along the
characteristics was essential in the integration by parts
above, and also that $g$ is even in $v$ respectively $w$.
The proof of Lemma~\ref{dis} is now complete.
\prfe
\smallskip
\noindent
{\em Regularization of $h\mathbf{1}_{\Omega _{m}}$}.\\
The function $h\mathbf{1}_{\Omega _{m}}$ is not smooth,
hence Lemma~\ref{ks} cannot be applied to it, and therefore
we smooth it. For fixed $m\in \mathbb N$ the function $h\mathbf{1}_{\Omega _{m}}$
is, as a function of $r,w,L$, supported in a set of the form
\begin{equation} \label{supphm}
Q_m := [R_0,R_1]\times [-W_0,W_0]\times [L_0,L_1]
\end{equation}
with $0<R_0<R_1$, $W_0>0$, and $0<L_0<L_1$; it will be important
that its support stays away both from $r=0$ and $L=0$.
Let $\zeta \in C_c^\infty (\R^3)$ be even in all three
variables, i.e.,
$\zeta(z_1,z_2,z_3) = \zeta(|z_1|,|z_2|,|z_3|)$, $\zeta \geq 0$,
$\int \zeta =1$, and define $\zeta_n :=n^3 \zeta(n\cdot)$ for $n\in \mathbb N$.
For $n$ sufficiently large we define
\[
h_n (r,w,L)
=\int_0^\infty \int_{-\infty}^\infty \int_0^\infty
(h\mathbf{1}_{\Omega _{m}})(\bar r,\bar w,\bar L)\,
\zeta _{n}(r-\bar r,w-\bar w,L-\bar L)\,d\bar L\,d\bar w\,d\bar r.
\]
Since $\Omega_m$ is strictly inside the set
$\{E\leq E_0\}$ with a positive distance from its boundary,
$h_n \in C^\infty_c(]0,\infty[\times \R \times ]0,\infty[)$,
and without loss of generality we can assume that
$\supp h_n \subset Q_m \cap \{E < E_0\}$.
Since $h$ is odd in $w$ respectively $v$ so is $h_n$, and clearly,
$h_n \to h \mathbf{1}_{\Omega _{m}}$ in $L^2\cap L^{1}$.
The crucial step is to show that
\begin{equation} \label{hnbracketlimit}
\lim_{n \to \infty }\{f_{0},h_n\}=g\mathbf{1}_{\Omega _{m}}
\end{equation}
in $L^2$. To this end we fix $(r,w,L)$, write
$\bar E :=E(\bar r,\bar w, \bar L)$ and
$\int:=\int_0^\infty \int_{-\infty}^\infty \int_0^\infty$ for brevity,
and split the convolution integral into three parts:
\begin{eqnarray*}
&&
\{f_{0},h_n\} =
-\phi_0'(E)
\left[w\partial_r +\left(\frac{L}{r^3}-U_0'(r)\right)\partial_w\right]\,h_n \\
&&
\quad = -\int
(h\mathbf{1}_{\Omega_{m}})(\bar r,\bar w,\bar L)(\phi_0'(E)-\phi_0'(\bar E))
\left[w\partial_r +\left(\frac{L}{r^3}-U_0'(r)\right)\partial_w\right]\\
&&
\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\zeta _{n}(r-\bar r,w-\bar w,L-\bar L)\,d\bar L\,d\bar w\,d\bar r\\
&&
\quad {} +
\int
(h\mathbf{1}_{\Omega _{m}})(\bar r,\bar w,\bar L) \phi_0'(\bar E)\,
\left[\bar w\partial_{\bar r} +\left(\frac{\bar L}{\bar r^3}-
U_0'(\bar r)\right)\partial_{\bar w}\right]\\
&&
\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\zeta _{n}(r-\bar r,w-\bar w,L-\bar L)\,d\bar L\,d\bar w\,d\bar r\\
&&
\quad {} -
\int
(h\mathbf{1}_{\Omega _{m}})(\bar r,\bar w,\bar L)\phi_0'(\bar E)\,
\left[(\bar w-w)\partial_{\bar r} +
\left(\frac{\bar L}{\bar r^3}-\frac{L}{r^3} -
U_0'(\bar r) + U_0'(r)\right)\partial_{\bar w}\right]\\
&&
\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\zeta _{n}(r-\bar r,w-\bar w,L-\bar L)\,d\bar L\,d\bar w\,d\bar r\\
&&
\quad =: - I_1 + I_2 - I_3 .
\end{eqnarray*}
By Lemma \ref{dis},
\[
I_2=
\int g\mathbf{1}_{\Omega _{m}}(\bar r,\bar w,\bar L)
\zeta _{n}(r-\bar r,w-\bar w,L-\bar L)\,d\bar L\,d\bar w\,d\bar r
\to g\mathbf{1}_{\Omega _{m}}\ \mbox{in}\ L^2.
\]
We show that $I_1$ and $I_3$ tend to zero in $L^2$. To do so we
change the integration variables into
$\tilde r = n (r-\bar r)$, $\tilde w = n (w-\bar w)$,
$\tilde L = n (L-\bar L)$ so that
$\bar r = r-\tilde r/n$, $\bar w = r-\tilde w/n$,
$\bar L = r-\tilde L/n$,
$d\bar L\,d\bar w\,d\bar r = n^{-3}d\tilde L\,d\tilde w\,d\tilde r$,
and $\partial_r \zeta_n(r-\bar r,w-\bar w,L-\bar L)=
n^4 \partial_{\tilde r} \zeta (\tilde r, \tilde w, \tilde L)$
with analogous formulas for the other derivatives. Thus
\begin{eqnarray*}
I_1
&=&
\int (h\mathbf{1}_{\Omega_{m}})(\bar r,\bar w,\bar L)
\frac{\phi_0'(E)-\phi_0'(\bar E)}{1/n}\\
&&
\quad
\left[w\partial_{\tilde r} +\left(\frac{L}{r^3}-U_0'(r)\right)
\partial_{\tilde w}\right]\,
\zeta (\tilde r, \tilde w, \tilde L)\,d\tilde L\,d\tilde w\,d\tilde r,\\
I_3
&=&
\int (h\mathbf{1}_{\Omega_{m}})(\bar r,\bar w,\bar L)\, \phi_0'(\bar E)\\
&&
\quad
\left[-\tilde w\partial_{\tilde r} +
\left(- \frac{\tilde L}{\bar r^3} + L \frac{\bar r ^{-3} - r^{-3}}{1/n}
+ \frac{U_0'(\bar r)-U_0'(r)}{1/n}\right)
\partial_{\tilde w}\right]\,
\zeta (\tilde r, \tilde w, \tilde L)\,d\tilde L\,d\tilde w\,d\tilde r
\end{eqnarray*}
Now
\[
\frac{\phi_0'(E)-\phi_0'(\bar E)}{1/n} \to
-\phi_0''(E)\,\left( w \tilde w + \frac{\tilde L}{2r^2} -
\frac{\tilde r L}{r^3} + \tilde r U_0'(r)\right),
\]
\[
\frac{\bar r ^{-3} - r^{-3}}{1/n} \to
\frac{3\tilde r}{r^4},\qquad
\frac{U_0'(\bar r)-U_0'(r)}{1/n} \to
- U_0''(r)\, \tilde r
\]
for $n \to \infty$, and all these limits are uniform with respect to
$(r,w,L) \in Q_m$ defined in (\ref{supphm}) and with respect
to $(\tilde r,\tilde w,\tilde L)\in \supp \zeta$.
Hence
\begin{eqnarray*}
&&
I_1 \to
- (h\mathbf{1}_{\Omega _{m}})(r,w,L)\phi_0''(E)\\
&&
\quad \int\left[ w \tilde w + \frac{\tilde L}{2r^2} -
\frac{\tilde r L}{r^3} + \tilde r U_0'(r)\right]
\left[w\partial_{\tilde r} +\left(\frac{L}{r^3}-U_0'(r)\right)
\partial_{\tilde w}\right]
\zeta (\tilde r, \tilde w, \tilde L)\,d\tilde L\,d\tilde w\,d\tilde r\\
&&
= 0,\\
&&
I_3 \to
(h\mathbf{1}_{\Omega _{m}})(r,w,L)\phi_0'(E)\\
&&
\quad \int\left[ -\tilde w \partial_{\tilde r} +
\left(-\frac{\tilde L}{\tilde r^3} +
\frac{3 L \tilde r}{r^4} - U_0''(r) \tilde r\right)\partial_{\tilde w}
\right]\,
\zeta (\tilde r, \tilde w, \tilde L)\,d\tilde L\,d\tilde w\,d\tilde r \\
&&
=0
\end{eqnarray*}
where both limits are in $L^2$ and the zeroes result from integration
by parts with respect to $\tilde r$ and $\tilde w$ and an exact cancellation.
The assertion (\ref{hnbracketlimit}) is now established.
\smallskip
\noindent
{\em Finally: The desired contradiction.}\\
The functions $h_n$
have all the properties required in Lemma~\ref{ks}, and we obtain
the estimate
\[
D^2 {{\cal H}_C} (f_{0})[\{f_{0},h_n\}]\geq -\frac{1}{2}
\int_{\{E<E_0\}} \phi_0'(E)\,
\frac{1}{r}\,U_0'(r)\, |h_n|^2\, dv\,dx.
\]
By (\ref{hnbracketlimit})
and the fact that $h_n$ and $g \mathbf{1}_{\Omega_{m}}$ are supported
in a common compact set,
$\nabla U_{\{f_0,h_n\}} \to \nabla U_{g\mathbf{1}_{\Omega _{m}}}$
in $L^2$ as $n\to \infty$. Since $\frac{1}{r}U_0'(r)$ is bounded on
$[0,\infty[$ we obtain the estimate
\[
D^2 {{\cal H}_C} (f_{0})[g\mathbf{1}_{\Omega _{m}}]
\geq -\frac{1}{2} \int_{\{E<E_0\}} \phi_0'(E)
\frac{1}{r}U_0'(r)\, h^2 \mathbf{1}_{\Omega _{m}}\, dv\,dx.
\]
Since $\lim_{m \to \infty }g\mathbf{1}_{\Omega _{m}}=g$ in $L^{1}\cap
L^2(\R^6)$, there exists $m_{0}$ sufficiently large such that
$g\mathbf{1}_{\Omega_{m_{0}}}\neq 0$ and by Lemma~\ref{dis},
$h\mathbf{1}_{\Omega _{m_{0}}}\neq 0$.
For all $m\geq m_{0}$, $\Omega _{m_{0}}\subset \Omega _{m}$ and
$h^2 \mathbf{1}_{\Omega _{m_{0}}}\leq h^2\mathbf{1}_{\Omega _{m}}$.
Since $\phi_0'(E)<0$ and $\frac{1}{r}U_0'(r)>0$ we have for all
$m\geq m_{0}$,
\[
D^2 {{\cal H}_C} (f_{0})[g\mathbf{1}_{\Omega _{m}}]
\geq -\frac{1}{2}\int_{\{E<E_0\}} \phi_0'(E)
\frac{1}{r}U_0'(r)\, h^2 \mathbf{1}_{\Omega _{m_0}}\, dv\,dx =: C >0.
\]
Hence $m \to \infty$ leads to a contradiction
to Lemma~\ref{reduction}, and the proof of Theorem~\ref{lower}
is complete.
\section{Appendix: Proof of Lemma~\ref{ks}}
\setcounter{equation}{0}
Let
\[
U_{h}(x) := \int\!\!\!\!\int \{f_{0},h\} dv\frac{dy}{|x-y|}
\]
be the potential induced by $-\{f_{0},h\}$, which is spherically symmetric.
A short computation using the definition (\ref{lie}) of the Lie-Poisson bracket
shows that
\[
\int \{f_{0},h\} dv = \nabla _{x}\cdot \int v\, \phi_0'(E)\, h(x,v)\,dv.
\]
By the spherical symmetry of $U_h$ and $h$,
\begin{eqnarray*}
U_{h}^{\prime }(r)
&=&
\frac{1}{r^2}\int_{|x|\leq r} \nabla _{x}\cdot \int v\, \phi_0'(E)\, h(x,v)\,dv\\
&=&
\frac{1}{r^2}\int_{|x|= r} \int \phi_0'(E)\, h(x,v) v\cdot\frac{x}{r} dv\, d\omega(x)
= 4 \pi \int w \phi_0'(E)\, h(x,v)\, dv.
\end{eqnarray*}
Therefore, by the Cauchy-Schwarz inequality,
\[
\frac{1}{8\pi }\int |\nabla _{x}U_{h}|^2dx
\leq
2\pi \int \left[ -\int w^2 \phi_0'(E)\,dv\right]\,
\left[ -\int \phi_0'(E)\, h^2 dv\right]\, dx.
\]
Since
\[
w^2 \phi_0'(E) = w \frac{d}{dw}\phi_0\left(\frac{1}{2} w^2 +\frac{L}{2 r^2}+U_0(r)\right)
\]
an integration by parts with respect to $w$ yields
\[
-\int w^2 \phi_0'(E)\,dv = \frac{\pi}{r^2}\int_0^\infty\int_{-\infty}^\infty
\phi_0\left(\frac{1}{2} w^2 +\frac{L}{2 r^2}+U_0(r)\right)\, dw\, dL = \rho_0(r).
\]
Hence
\[
D^2 {{\cal H}_C} (f_{0})[\{f_{0},h\}]
\geq
-\frac{1}{2} \int\!\!\!\!\int
\phi_0' (E)\,\left(|\{E,h\}|^2 - 4\pi \rho_{0} h^2\right)\,dv\,dx.
\]
Since $h$ is odd in $v$ respectively $w$ the function
\[
\mu(r,w,L)
:= \frac{1}{r w} h(r,w,L)
\]
is smooth away from $r=0$; in passing we notice that the functions $h_n$
to which we applied Lemma~\ref{ks} in the proof of Theorem~\ref{lower}
have support bounded away from $r=0$, but this is not necessary for Lemma~\ref{ks}.
Since $h=r w \mu$,
\[
\{E,h\} = rw \{E,\mu \} + \mu \{E,rw\},
\]
and hence
\begin{eqnarray*}
|\{E,h\}|^2 &=&(rw)^2|\{E,\mu \}|^2+rw\{E,rw\}\{E,\mu
^2\}+\mu ^2|\{E,rw\}|^2 \\
&=&
(rw)^2|\{E,\mu \}|^2 + \{E,\mu ^2rw\{E,rw\}\}-\mu^2rw\{E,\{E,rw\}\}.
\end{eqnarray*}
The first term is as claimed in Lemma~\ref{ks}.
The second term leads to
$\{f_{0},\mu ^2rw\{E,rw\}\}$ whose integral with respect to $x$ and $v$
vanishes; if we cut a small ball of radius $\epsilon$
about $x=0$ from the $x$-integral
then the surface integral appearing after the integration by parts
with respect to $x$ vanishes for $\epsilon \to 0$ since $r \mu^2 \leq C/r$.
By the Poisson equation,
\[
\{E,\{E,rw\}\}=-2 w U_0' - w (r U_0')'=- r w\, \left(4\pi \rho_{0} + \frac{1}{r}U_0'\right).
\]
Hence the third term above becomes
$ 4\pi \rho _{0}h^2 + h^2 \frac{1}{r}U_0'$, and Lemma~\ref{ks} is proven.
\prfe
\noindent
{\bf Acknowledgment.}
The research of the first author
is supported in part by an NSF grant. This article is dedicated
to the memory of Xudong Liu.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,708 |
\section{Introduction}
The time evolution of an electromagnetic field in free space is described by Maxwell's equations in $\RR^3$:
\begin{alignat}{2}
&\partial_t E=\curl B\,, \qquad &&\partial_t B=-\curl E\\
&\Div E=0\,, \qquad &&\Div B=0\,,
\end{alignat}
where we have set all constants equal to~$1$. Here, $E$ and $B$ are time-dependent vector fields that stand for the electric and magnetic part of the electromagnetic field. As usual, $\curl$ and $\Div$ denote the rotor and divergence of a vector field in $\RR^3$, respectively.
The integral curves of the electric field at fixed time $t$, i.e., the solutions to the autonomous ODE
\begin{align*}
\frac{dx(\tau)}{d\tau}=E(x(\tau),t)
\end{align*}
for some initial condition $x(0)=x_0$, are the \emph{electric lines} at time $t$. Analogously, the \emph{magnetic lines} are the integral curves of the magnetic field at each fixed time~$t$. In general, the electric and magnetic lines change their topology and bifurcate during the time evolution~\cite{ABT17}. However, for a particularly relevant class of solutions to Maxwell's equations, the so-called \emph{null solutions}, the electric and magnetic lines preserve their topology for all time.
\subsection{Stable electromagnetic structures}\label{S.null} We recall that an electromagnetic field $E(x,t),B(x,t)$ is a null solution to Maxwell's equations, if the electric and magnetic fields are orthogonal and have the same modulus, i.e., $E\cdot B=0$ and $|E|=|B|$, for all $t$. Then the time evolution of the electric and magnetic lines is given by a $1$-parameter family of diffeomorphisms $\Phi_t:\mathbb{R}^3\to\mathbb{R}^3$, where $\Phi_0$ is equal to the identity. More precisely, if the normalized Poynting field
\[
P:=\frac{E\times B}{W}
\]
is smooth everywhere, with $W:=\frac12(E^2+B^2)$ the energy density, it follows~\cite{Newcomb,kbbpsi,kpsi} that $E$ and $B$ can be written in terms of the initial electric and magnetic fields $E_0,B_0$ as
\begin{align*}
E(t)=h(t)\Phi_{t*}E_0\,,\qquad B(t)=h(t)\Phi_{t*}B_0\,,
\end{align*}
where $\Phi_{t*}$ is the push-forward of the non-autonomous flow defined by $P$, and $h$ is an explicit (in terms of $W$) non-vanishing function whose expression is not relevant for our purposes. A striking conclusion of this property is that any topological electromagnetic structure, such as knotted closed field lines or invariant tori, that $E$ or $B$ may contain at time $t=0$ is preserved for all time if the corresponding solution to Maxwell's equations is null. We shall then say that the topological structures of the initial electromagnetic field are \emph{stable} under time evolution.
In view of the previous discussion, the study of stable topological structures in electromagnetic fields is thus strongly connected to the study of null solutions to Maxwell's equations. A remarkable construction of such null fields is due to Bateman~\cite{bateman}. Roughly speaking, Bateman showed that given a map $(\alpha,\beta):\mathbb{R}^{3}\times\RR\to\mathbb{C}^2$, referred to as a pair of \textit{Bateman variables}, satisfying a certain PDE and any holomorphic function $h:\mathbb{C}^2\to\mathbb{C}$ there is a recipe to obtain a null electromagnetic field; the details of this construction will be reviewed in Section~\ref{sec:bateman}. A particular choice of the functions $(\alpha,\beta)$ and $h$ yields a solution to Maxwell's equations that is called the \emph{Hopfion}, which is intimately linked to the Hopf fibration on the 3-sphere $\SS^3$. In this solution any field line is closed and any pair of field lines forms a Hopf link, for all time. This important solution was first found by Synge~\cite{synge}, but its topological content was unveiled by Ra{\~n}ada in~\cite{R,R2} following a totally different method. It has been object of intense research during the last $25$ years, including a proposal for experimental realization~\cite{cameron}. Generalizations of the Hopfion were obtained in~\cite{kbbpsi}, where the authors constructed null solutions (following Bateman's method) with electric and magnetic lines in the shape of arbitrary torus knots. These solutions are of \textit{Hopf type}, i.e., they are constructed with the same choice of $(\alpha,\beta)$ as the Hopfion, but with different choices of the holomorphic function $h$. The interested reader may consult~\cite{ABT17} for a comprehensive account on the subject. As an aside remark, we mention that in~\cite{kpsi} the initial electromagnetic fields $E_0,B_0$ that yield null solutions under the evolution of Maxwell's equations were characterized. However it is very hard to use such a characterization effectively, so Bateman's construction remains as the only effective technique to construct null electromagnetic fields.
\subsection{A connection with contact geometry}
In spite of the clarity of Bateman's method, the construction of null electromagnetic fields encoding knots and links that are not of torus type is very challenging, and has been solved only recently~\cite{bode}. The key observation to achieve this result is a surprising connection between null solutions and contact geometry. Specifically, it was proved that any Bateman field of Hopf type is related to a pair of \emph{Legendrian vector fields} with respect to the standard contact structure on $\SS^3$ given by the kernel of the standard contact 1-form
\begin{equation}\label{eq:std}
-y_1\dd x_1+x_1\dd y_1-y_2\dd x_2+x_2 \dd y_2,
\end{equation}
where we use $z_i=x_i+\I y_i$, $i=1,2$, as coordinates on $\mathbb{C}^2$. These two Legendrian fields are real multiples of pushforwards of the electric and the magnetic field on $\RR^3$ at time $t=0$ by the Bateman variables $(\alpha,\beta)|_{t=0}$. Throughout this paper, $\SS^3$ will be defined as the unit sphere in $\CC^2\cong \RR^4$, i.e., $\SS^3=\{(z_1,z_2)\in\CC^2:|z_1|^2+|z_2|^2=1\}$, and is endowed with the standard volume form, which is precisely the volume compatible with the standard contact form~\eqref{eq:std}.
In contrast with the usual flexibility in contact geometry, we emphasize that, in this work, isotopies of the standard contact structure are not allowed, unless they leave the kernel of the standard contact form invariant. Accordingly, by Legendrian we shall always mean an object (vector field, curve, ...) that is tangent to the kernel of the $1$-form~\eqref{eq:std}.
The goal of this article is to develop and exploit the aforementioned relation between stable electromagnetic structures (via null solutions) and Legendrian vector fields with respect to the standard contact structure. This connection is beneficial in both directions. On the one hand, it allows us to obtain electromagnetic Bateman solutions by studying Legendrian vector fields on $\SS^3$; conversely, it provides a way to construct Legendrian fields with prescribed topological structures, such as knotted periodic orbits or invariant tori, using Bateman's method. As an aside remark, we observe that Arnold in~\cite[Problem 1994-13]{Arnold} considered the problem of studying periodic orbits of Legendrian vector fields. Specifically:
\emph{Consider a particle in a magnetic field on a surface $M^2$. Study Legendrian divergence-free vector fields on $ST^*M^2$ and, in particular, their closed orbits. More generally, consider divergence-free Legendrian vector fields on $\mathbb S^3$ for some (standard?) contact structure. Does there exist a counterexample to the Seifert conjecture in this class of vector fields?}
Although we do not obtain any insight on the motion of a charged particle in a magnetic field on a surface (see Remark~\ref{R:charge}), we hope the new techniques we introduce in this work may be useful for related problems on divergence-free Legendrian fields.
\subsection{Statements of the theorems}\label{SS.state}
Our first main result is a characterization of those Legendrian vector fields on $\SS^3$ that yield null electromagnetic fields on $\RR^3$, via Bateman's construction. As we shall review in Section~\ref{sec:bateman}, the vector fields on $\RR^3$ are obtained from those on $\SS^3$ using the inverse projection map $(\alpha,\beta)|_{t=0}^{-1}$, which is a composition of the stereographic projection and a mirror reflection. The complex structure $J:T\mathbb{C}^2\to T\mathbb{C}^2$ maps any given vector field $X$ that is Legendrian with respect to the standard contact structure on $\SS^3$ to the vector field $JX$, which is the unique Legendrian vector field on $\SS^3$ (with respect to the standard contact structure) such that $|X|=|JX|$, $X\cdot JX=0$ and $(X,JX)$ is positively oriented. Here, the vector product and norm are computed using the canonical metric on $\SS^3$.
\begin{theorem}\label{P:m1}
Let $E_0$ be a real-analytic vector field on $\SS^3$ that is Legendrian with respect to the standard contact structure. Then $E_0$ and $B_0:=JE_0$ yield an electromagnetic Bateman solution on $\RR^3$ associated with the Hopf fibration if and only if $E_0$ and $B_0$ are divergence-free.
\end{theorem}
Next, using Bateman's technique, we construct non-vanishing divergence-free Legendrian vector fields with a prescribed set of periodic orbits of vanishing rotation numbers (details are provided in Section~\ref{sec:kpo}); see the aforementioned Arnold's problem for an additional motivation to study divergence-free Legendrian fields.
\begin{theorem}\label{P:m2}
Let $X$ be a non-vanishing Legendrian field on $\SS^3$ (with respect to the standard contact form). Then any periodic orbit of $X$ has vanishing rotation number. Conversely, given any Legendrian link $L=\cup_{i=1}^n L_i$ in $\SS^3$, where each $L_i$ has vanishing rotation number, there is a non-vanishing, real-analytic, divergence-free Legendrian field on $\SS^3$ (with respect to the standard contact structure) such that $L$ is isotopic to a subset of its periodic orbits.
\end{theorem}
\begin{remark}
The only condition on each $L_i$ is the vanishing rotation number, so that every Legendrian knot, no matter which value its Thurston-Bennequin invariant takes, can be realized as a periodic orbit. Note in particular, that every link type has a Legendrian representative, all of whose components have vanishing rotation number, so these assumptions do not restrict the topology of the link.
\end{remark}
As emphasized, the proof of Theorem~\ref{P:m2} makes use of Bateman's construction. Accordingly, we also obtain null solutions to Maxwell's equations with stable closed electromagnetic lines, which allows us to extend~\cite[Theorem~1]{bode} to the context of non-vanishing electromagnetic fields.
\begin{corollary}\label{cor:rot}
For every link $L=\cup_{i=1}^n L_i$ in $\RR^3$ and every (possibly empty or non-proper) subset $I\subset\{1,2,\cdots,n\}$, there is a smooth isotopy of diffeomorphisms $\Phi_t:\RR^3\to\RR^3$ and a non-vanishing electromagnetic field such that $\Phi_t(\cup_{i\in I}L_i)$ is a subset of the closed electric lines and $\Phi_t(\cup_{i\notin I}L_i)$ is a subset of the closed magnetic lines for all $t\in\mathbb{R}$.
\end{corollary}
In general, the electric and magnetic lines of the solutions obtained in Corollary~\ref{cor:rot} are not all closed. In fact, the existence of the Hopfion solution~\cite{R,R2}, where all field lines are closed, motivates the question of whether Seifert foliations of the 3-sphere other than the Hopf fibration can be realized by Bateman fields or not. In this direction, we can show that any (positive) Seifert foliation is diffeomorphic to a Legendrian foliation, with respect to the standard contact structure. In view of the contact geometry connection mentioned before, this is a first step to construct stable electromagnetic fields diffeomorphic to Seifert foliations.
\begin{proposition}\label{P:seifert}
Any Seifert foliation $S_{p,q}$, $p\geq1$, $q\geq1$, of $\SS^3$ is isotopic to the foliation defined by a non-vanishing, real-analytic, divergence-free Legendrian vector field $X$ with respect to the standard contact structure.
\end{proposition}
\begin{remark}
In view of Theorem~\ref{P:m1}, the only point that prevents us from concluding that $X$ is of Bateman-type is that we cannot check that $JX$ is divergence-free.
\end{remark}
The fields constructed in \cite{kbbpsi} have the property that they contain a set of closed field lines in the shape of a positive torus link. In that construction different torus links lead to different Bateman fields. In our last main result we show that there is a single Bateman field that contains every positive torus link. To prove this claim we employ a symmetry of Bateman's construction that allows us to obtain a new pair of Bateman variables $(\tilde{\alpha},\tilde{\beta})$ from a known pair of Bateman variables $(\alpha,\beta)$ by applying complex conjugation and time reversal.
\begin{proposition}\label{P:torus}
There exists a null electromagnetic field whose electric part at time $t=0$ (and hence at any other $t\in\RR$) contains every positive torus knot as a periodic orbit.
\end{proposition}
\subsection{Organization of the paper}
The remainder of this paper is structured as follows. Section~\ref{sec:bateman} reviews Bateman's construction and its relation to the Hopf fibration and Legendrian fields on $\SS^3$. This machinery is used in Section~\ref{sec:teo1} to prove Theorem~\ref{P:m1}. Section~\ref{sec:kpo} concerns periodic orbits of Legendrian vector fields. Theorem~\ref{P:m2} is then proved in Section~\ref{sec:teo2}, while the discussion on Legendrian Seifert foliations and the proof of Proposition~\ref{P:seifert} is presented in Section~\ref{sec:seifert}. We investigate transformations between Bateman variables via complex conjugation and time reversal and prove Proposition~\ref{P:torus} in Section~\ref{sec:torus}. Finally, in Section~\ref{sec:int} we make several observations about integrable Bateman fields that are of independent interest and provide an explanation of the integrability properties of the fields constructed in~\cite{kbbpsi} and a connection with Rudolph's totally tangential $\mathbb{C}$-links~\cite{rudolphtt,rudolphtt2}.
\section{Bateman's construction of null electromagnetic fields}\label{sec:bateman}
In this section we describe Bateman's construction, which associates to a holomorphic function a null electromagnetic field, and its surprising connection with contact geometry.
As usual, in this work we shall represent an electromagnetic field by its Riemann-Silberstein vector
$$F=E+\I B:\mathbb{R}^{3+1}\to\mathbb{C}^3\,,$$
whose real part $E$ is the (time-dependent) electric field and whose imaginary part $B$ is the magnetic field. For an overview of the history and many uses of the Riemann-Silberstein formulation we point the reader to~\cite{RS}.
Let $\alpha, \beta:\mathbb{R}^{3+1}\to\mathbb{C}$ be two functions satisfying the PDE
\begin{equation}
\label{eq:1}
\nabla \alpha\times\nabla\beta=\I (\partial_t\alpha\nabla\beta-\partial_t\beta\nabla\alpha)\,,
\end{equation}
where $\nabla$ denotes the gradient operator with respect to the three spatial variables.
Bateman showed in~\cite{bateman} that every pair $f,g:\mathbb{C}^2\to\mathbb{C}$ of holomorphic functions defines an electromagnetic field
\begin{equation*}
F=E+\I B=\nabla f(\alpha,\beta)\times\nabla g(\alpha,\beta)\,,
\end{equation*}
which is null for all time $t$.
Using $(z_1,z_2)$ as coordinates of $\mathbb{C}^2$ and setting $h:=\partial_{z_1} f\partial_{z_2} g-\partial_{z_2} f\partial_{z_1} g$, we can rewrite $F$ as
\begin{equation}
\label{eq:bateman}
F=h(\alpha, \beta)\nabla\alpha\times\nabla\beta\,.
\end{equation}
Since $f$ and $g$ are arbitrary holomorphic functions, any holomorphic function $h$ can be used to define a Bateman field. Conversely, any (real-analytic) null field can be locally obtained via Bateman's construction~\cite{hogan} with some choice of $(\alpha,\beta)$ and holomorphic $h$. Actually, it suffices that $h$ be holomorphic on a neighbourhood of the image of $(\alpha,\beta)$.
Since these solutions to Maxwell's equations are null, the topology of their field lines is preserved for all time. That is, as recalled in Section~\ref{S.null}, the time evolution of a field line is given by transporting it along the normalized Poynting vector field.
The main difficulty in Bateman's construction is to find global solutions to Equation~\eqref{eq:1}. A particularly interesting and fruitful choice of $(\alpha,\beta)$ is (see e.g.~\cite{kbbpsi}):
\begin{align}
\label{eq:stereo3}
\alpha(x,y,z,t)&=\frac{x^2+y^2+z^2-t^2-1+2\I z}{x^2+y^2+z^2-(t-\I)^2}\,,\nonumber\\
\beta(x,y,z,t)&=\frac{2(x-\I y)}{x^2+y^2+z^2-(t-\I)^2}\,.
\end{align}
It is easy to check that these functions solve the PDE~\eqref{eq:1}. Notice that
$$|\alpha|^2+|\beta|^2=1\,,$$ so that the image of the map $(\alpha,\beta)$ lies in the $3$-sphere. In fact, it was shown in~\cite{bode} that at $t=0$ the map $(\alpha,\beta)$ is the inverse of the stereographic projection $\SS^3\to\mathbb{R}^3\cup\{\infty\}$ followed by a mirror reflection. More precisely, denoting the composition of the stereographic projection and the mirror reflection along the plane $\{y=0\}$ by $\phi_0:\SS^3\to\mathbb{R}^3\cup\{\infty\}$, we have $\phi_0(\alpha,\beta)|_{t=0}=\id$. In fact, for any value of $t$ the map $(\alpha,\beta)$ defines a diffeomorphism from $\mathbb{R}^3\cup\{\infty\}\to\SS^3$.
It is clear that the Poynting field of a Bateman solution does not depend on the holomorphic function $h$. In the particular case of the solution $(\alpha,\beta)$ presented in Equation~\eqref{eq:stereo3}, at time $t=0$ the Poynting field is (up to a real factor) the pushforward by $\phi_0$ of the Hopf field on $\SS^3$
\begin{equation}\label{eq:hopf}
-y_1\partial_{x_1}+x_1\partial_{y_1}-y_2\partial_{x_2}+x_2\partial_{y_2}\,.
\end{equation}
Its integral curves are tangent to the fibers of the Hopf fibration, so that all field lines are closed and any pair of them forms a Hopf link (identifying $\RR^3\cup\{\infty\}$ with $\SS^3$). In what follows, a Bateman solution with the choice of $(\alpha,\beta)$ as in Equation~\eqref{eq:stereo3} will be called \emph{of Hopf type}. Note that the Hopf field~\eqref{eq:hopf} is the dual of the standard contact form on $\SS^3$, cf. Equation~\eqref{eq:std}. It then follows that at time $t=0$ both the electric and magnetic part of any Bateman field of Hopf type are pushforwards by $\phi_0$ of Legendrian fields on $\SS^3$ (with respect to the standard contact form).
The time evolution of these fields can then be understood as keeping the Legendrian fields on $\SS^3$ fixed, but varying the projection map $(\alpha,\beta)$ (or rather $(\alpha,\beta)|_{t=t_*}^{-1}:\SS^3\backslash\{(1,0)\}\to\mathbb{R}^3$) via its $t$-dependence. More precisely, notice that the fields
\begin{align}\label{eq:v1v2}
v_1&=-x_2\partial_{x_1}+y_2\partial_{y_1}+x_1\partial_{x_2}-y_1\partial_{y_2}\,,\nonumber\\
v_2&=-y_2\partial_{x_1}-x_2\partial_{y_1}+y_1\partial_{x_2}+x_1\partial_{y_2}\,,
\end{align}
form a basis of the standard contact plane at each point in $\SS^3$. They are isometric (via rotations) to the Hopf field in Equation~\eqref{eq:hopf}, so all their field lines are closed. It turns out that, for each fixed time $t$, they are (up to the same real factor, given explicitly in~\cite{bode}) the pushforwards of $\text{Re}(\nabla\alpha\times\nabla\beta)$ and $\text{Im}(\nabla\alpha\times\nabla\beta)$ by the map $(\alpha,\beta)|_t$. In other words, they correspond to the electric part of a Bateman solution of Hopf type with $h=1$ and $h=-\I$, respectively.
Therefore, the field
\begin{equation}
\widetilde{F}:=h(z_1,z_2)(v_1+\I v_2)
\end{equation}
is the pushforward by $(\alpha,\beta)|_t$ of the Bateman field in Equation~\eqref{eq:bateman} with the choice~\eqref{eq:stereo3}, up to a time dependent real factor. In particular, the two fields $F$ and $\widetilde{F}$ have topologically identical field lines for all $t$, and one of them is non-vanishing if and only if the other is non-vanishing. We say that the real and imaginary part of $\widetilde{F}$ form \textit{the pair of Legendrian fields corresponding to the Bateman solution} $F$ in Equation~\eqref{eq:bateman}. Note that $\widetilde{F}$ is defined on $\SS^3$, while $F$ is defined on $\mathbb{R}^{3+1}$. We also say that a pair $(E_0,B_0)$ of Legendrian fields on $\SS^3$ is of Bateman-type if it arises as $\widetilde{F}=E_0+\I B_0$ for some Bateman solution $F$ of Hopf type.
\begin{remark}\label{rem}
As observed in \cite{bode}, the fact that the inverse of $(\alpha,\beta)|_{t=0}$ is not the stereographic projection map, but rather the composition of a stereographic projection with a mirror reflection, means that if a Bateman type Legendrian field on $\SS^3$ has a closed orbit of knot type $K$, then the corresponding electric or magnetic part in $\RR^3$ has a closed orbit, whose knot type is the mirror image of $K$. For example, the electromagnetic fields in~\cite{kbbpsi} possess field lines in the shape of positive torus knots in $\mathbb{R}^3$, but the corresponding field lines in the fields on $\SS^3$ are negative torus knots.
\end{remark}
\section{From Legendrian fields to Bateman solutions: proof of Theorem~\ref{P:m1}}\label{sec:teo1}
Let $F$ be a Bateman solution to Maxwell's equations in $\RR^{3+1}$ of Hopf type. As presented in Section~\ref{sec:bateman} (see~\cite{bode} for details), it yields a Bateman-type pair of Legendrian fields $(E_0,B_0) $ on $\SS^3$ with respect to the standard contact structure. By construction, these fields are real-analytic, divergence-free and $B_0=JE_0$ (recall the definition of $J$ in Section~\ref{SS.state}). In this section we show that, conversely, every pair of Legendrian fields with these properties is of Bateman type. As an application, we obtain a simple topological necessary condition for a vector field to be the electric or magnetic part of a Bateman solution of Hopf type.
First, we need some background on the kind of functions $\Th:\SS^3\to\mathbb{C}$ that can be extended to holomorphic functions on a neighbourhood of $\SS^3$. Let $M$ be a smooth real submanifold of $\mathbb{C}^n$. Its complex tangent space $H_pM$ at a point $p\in M$ is defined as $T_pM\cap J(T_pM)$, where $J$ is the complex structure on $\mathbb{C}^n$. $M$ is said to be a \textit{CR-manifold} if the complex dimension of $H_pM$ does not depend on $p\in M$.
The 3-sphere $\SS^3$ is an example of a CR-manifold in $\mathbb{C}^2$, where the complex tangent space is of (complex) dimension 1, given precisely by the contact distribution of the standard contact form, that is, at each point in $\SS^3$ it is generated by $v_1$ and $v_2$ introduced in Equation~\eqref{eq:v1v2}. We can think of the complex tangent space as a complex line bundle over $\SS^3$ generated by
\begin{equation}
\mathbb{L}=(-x_2+\I y_2)\frac{\partial}{\partial z_1}+(x_1-\I y_1)\frac{\partial}{\partial z_2}\,,
\end{equation}
with $\tfrac{\partial}{\partial z_i}=\tfrac{1}{2}(\tfrac{\partial}{\partial x_i}-\I\tfrac{\partial}{\partial y_i})$.
Note that $\text{Re}(\mathbb{L})=\tfrac{1}{2}v_1$ and $\text{Im}(\mathbb{L})=\tfrac{1}{2}v_2$. The \textit{tangential Cauchy-Riemann operator} is then defined as
\begin{equation}
\overline{\mathbb{L}}=(-x_2-\I y_2)\frac{\partial}{\partial\overline{z_1}}+(x_1+\I y_1)\frac{\partial}{\partial \overline{z_2}}
\end{equation}
with $\tfrac{\partial}{\partial \overline{z_i}}=\tfrac{1}{2}(\tfrac{\partial}{\partial x_i}+\I\tfrac{\partial}{\partial y_i})$.
A smooth function $f:\SS^3\to\mathbb{C}$ is called a \textit{CR-function} if it satisfies the \textit{tangential Cauchy-Riemann equations}:
\begin{equation}
\overline{\mathbb{L}}f=0\,.
\end{equation}
CR-functions play an important role in the context of holomorphic extensions of functions on submanifolds of complex space. If $\tilde{f}:\SS^3\to\mathbb{C}$ is a real-analytic function on $\SS^3$, then $\tilde{f}$ satisfies the tangential Cauchy-Riemann equations if and only if $\tilde{f}$ extends to a holomorphic function on some neighbourhood of $\SS^3$, i.e., there is a neighbourhood $U\subset \CC^2$ of $\SS^3$ and a holomorphic function $f:U\to\mathbb{C}$ such that $f|_{\SS^3}=\tilde{f}$. This result (with varying degree of generality) is attributed to Severi~\cite{severi1931, severi1958} and Tomassini~\cite{tomassini}, and holds for any real-analytic CR-manifold and its corresponding system of tangential Cauchy-Riemann equations. See~\cite{CR} for a recent account on the subject.
Now let us consider the Legendrian fields from the statement of Theorem~\ref{P:m1}. Since $E_0$ and $B_0=JE_0$ are Legendrian and real-analytic, it easily follows that they can be expressed as linear combinations of $v_1$ and $v_2$ (cf. Equation~\eqref{eq:v1v2}):
\begin{align*}
E_0&=\text{Re}(\Th)v_1-\text{Im}(\Th)v_2\,,\nonumber\\
B_0&=\text{Re}(\Th)v_2+\text{Im}(\Th)v_1\,,
\end{align*}
for some real-analytic function $\Th:\SS^3\to\mathbb{C}$. We compute the divergence of $E_0$ and $B_0$ and find that
\begin{align}\label{eq.div}
\text{div}(E_0)=& \text{ div}\Big(\text{Re}(\Th)v_1-\text{Im}(\Th)v_2\Big)\nonumber\\
=&-x_2\frac{\partial \text{Re}(\Th)}{\partial x_1}+y_2\frac{\partial \text{Im}(\Th)}{\partial x_1}+y_2\frac{\partial \text{Re}(\Th)}{\partial y_1}+x_2\frac{\partial \text{Im}(\Th)}{\partial y_1}\nonumber\\
&+x_1\frac{\partial \text{Re}(\Th)}{\partial x_2}-y_1 \frac{\partial \text{Im}(\Th)}{\partial x_2}-y_1\frac{\partial \text{Re}(\Th)}{\partial y_2}-x_1\frac{\partial\text{Im}(\Th)}{\partial y_2}\nonumber\\
=&\text{ Re}(\mathbb{L}(\Th))\,.
\end{align}
Similarly, we have
\begin{align}
\text{div}(B_0)=&\text{ div}\Big(\text{Re}(\Th)v_2+\text{Im}(\Th)v_1\Big)\nonumber\\
=&-x_2\frac{\partial \text{Im}(\Th)}{\partial x_1}-y_2\frac{\partial \text{Re}(\Th)}{\partial x_1}+y_2\frac{\partial \text{Im}(\Th)}{\partial y_1}-x_2\frac{\partial \text{Re}(\Th}{\partial y_1}\nonumber\\
&+x_1\frac{\partial \text{Im}(\Th)}{\partial x_2}+y_1\frac{\partial \text{Re}(\Th)}{\partial x_2}-y_1 \frac{\partial \text{Im}(\Th)}{\partial y_2} +x_1\frac{\partial \text{Re}(\Th)}{\partial y_2}\nonumber\\
=&\text{ Im}(\mathbb{L}(\Th)).
\end{align}
Therefore, $E_0$ and $B_0$ are both divergence-free if and only if the analytic function $\Th$ satisfies the tangential Cauchy-Riemann equations. In turn, this is equivalent to the existence of a holomorphic extension of $\Th$ on a neighbourhood of $\SS^3\subset\CC^2$. Accordingly, there is a neighbourhood $U\subset \CC^2$ of $\SS^3$ and a holomorphic function $h:U\to\mathbb{C}$ such that
\begin{equation}\label{eq:rest}
h|_{\SS^3}=\Th\,.
\end{equation}
Finally, taking $(\alpha,\beta):\RR^{3+1}\to\CC^2$ as in Equation~\eqref{eq:stereo3}, we notice that Bateman's construction works if the function $h$ is holomorphic in a neighborhood of the image of $(\alpha,\beta)$, which is $\SS^3\subset\CC^2$. Accordingly, the Bateman field
\[
F=h(\alpha,\beta)(\nabla\alpha\times\nabla\beta)
\]
is a null solution to Maxwell's equations in $\RR^{3+1}$. Since $h$ satisfies Equation~\eqref{eq:rest}, $(E_0,B_0)$ is the pair of Legendrian fields on $\SS^3$ corresponding to the solution $F$. This completes the proof of the theorem.
\begin{remark}
Since the Legendrian fields of Bateman type are (up to a known real factor) pushforwards by the map $(\alpha,\beta)|_{t=0}$ of the initial electric and magnetic fields, we can interpret Theorem~\ref{P:m1} as a result comparable to~\cite[Section~3]{kpsi}, in that it provides a possibility to investigate if an electromagnetic field is null for all time by studying only its initial datum at $t=0$.
\end{remark}
\begin{remark}\label{R:charge}
There is an interesting connection between the study of divergence-free Legendrian fields and the magnetic geodesic problem~\cite{Ginz}. The magnetic geodesic problem on $\SS^2$ yields a Hamiltonian vector field on $\SS^3$ which is Legendrian with respect to the standard contact form, and divergence-free. Unfortunately, it cannot be embedded into a Bateman-type couple as in Theorem~\ref{P:m1} unless the magnetic field is identically zero on $\SS^2$. In this case one obtains a Hopf field on $\SS^3$.
\end{remark}
To conclude this section, we show that any Bateman-type non-vanishing field on $\SS^3$ is homotopic to the Hopf field $v_1$ in Equation~\eqref{eq:v1v2} via non-vanishing Legendrian fields. Since any Bateman-type field is Legendrian with respect to the standard contact structure, the claim follows from the following standard result (we include a short proof for the sake of completeness).
\begin{proposition}\label{P:m3}
Let $X$ be a non-vanishing Legendrian field on $\SS^3$ (with respect to the standard contact structure). Then $X$ is homotopic to $v_1$ via non-vanishing Legendrian fields.
\end{proposition}
\begin{proof}
As in the proof of Theorem~\ref{P:m1}, we can represent every Legendrian field by a complex-valued function $h$ on $\SS^3$, which expresses the field in the basis $\{v_1,v_2\}$. This identification provides a homeomorphism between the space of non-vanishing Legendrian fields with respect to the standard contact structure and the space of non-vanishing complex-valued functions $h:\SS^3\to\mathbb{C}\backslash\{0\}$. Since $\pi_3(\mathbb{C}\backslash\{0\})=0$, any two such maps are homotopic, which implies that the corresponding Legendrian fields are homotopic via non-vanishing Legendrian fields. In particular, $X$ is homotopic to $v_1$, which corresponds to the function $h=1$.
\end{proof}
\section{Knotted periodic orbits of non-vanishing Legendrian fields}\label{sec:kpo}
In this section we focus on non-vanishing Legendrian fields (with respect to the standard contact structure) and their periodic orbits. As explained in Section~\ref{sec:bateman}, a periodic orbit of the electric (or magnetic) part of a Bateman solution of Hopf type (at $t=0$) corresponds to a periodic orbit of the associated Legendrian field on $\SS^3$. It is thus a Legendrian knot with respect to the standard contact structure. In Theorem~\ref{P:m2} we claim that whether or not a Legendrian knot can be realized by a periodic orbit of a non-vanishing Legendrian field (and of a non-vanishing Bateman field) is only determined by one of the classical contact invariants of Legendrian knots, the rotation number~\cite[Section~3.5]{Geiges}:
\begin{definition}\label{def:rot}
Let $K$ be a null-homologous Legendrian knot in a contact manifold $(M,\xi)$ and let $\Sigma$ be an embedded orientable surface with $\partial \Sigma=K$. Given a trivialization of the restriction of the contact bundle $\xi|_{\Sigma}$ and a non-vanishing tangent vector field $V$ on $K$, we can interpret $V$ as a loop in $\mathbb{C}\backslash\{0\}$ via its coordinates with respect to the given trivialization. Then the \emph{rotation number} $\cR(K)$ is defined as the winding number of this loop, and is independent of the trivialization.
\end{definition}
\begin{remark}
We recall that any orientable $2$-plane bundle over a surface with boundary is trivial, so the trivialization in Definition~\ref{def:rot} always exists. In the case of the standard contact structure on $\SS^3$, the vector fields $\{v_1,v_2\}$ introduced in Equation~\eqref{eq:v1v2} provide a basis of such a trivialization, which is, in fact, a global basis. Accordingly, the rotation number of a Legendrian knot on $\SS^3$ (with respect to the standard contact structure) is independent of the Seifert surface $\Sigma$.
\end{remark}
In Section~\ref{sec:teo2} we prove Theorem~\ref{P:m2} and Corollary~\ref{cor:rot} using Bateman's construction and the correspondence explained in Section~\ref{sec:bateman} between Legendrian fields and null solutions of Maxwell's equations. The problem of realizing any foliation by circles on $\SS^3$ as a Bateman-type electric (or magnetic) field is explored in Section~\ref{sec:seifert}, where we prove Proposition~\ref{P:seifert}.
\subsection{Proof of Theorem~\ref{P:m2} and Corollary~\ref{cor:rot}}\label{sec:teo2}
Let $X$ be a Legendrian (with respect to the standard contact structure) and non-vanishing vector field on $\SS^3$. We argued in Section~\ref{sec:teo1} that it can be written in terms of the Hopf basis $\{v_1,v_2\}$ as
\begin{equation}
\text{Re}(h(v_1+\I v_2))
\end{equation}
for some complex-valued function $h:\SS^3\to\mathbb{C}\backslash\{0\}$. Obviously, the function $h$ is defined by expressing $X$ in terms of the aforementioned basis of the contact distribution.
Let $K$ be a periodic orbit of~$X$. Since $\SS^3$ is simply-connected, $K$ is null-homotopic and hence there is a homotopy of loops $K_s$, $s\in[0,1]$, with $K_0=K$ and $K_1$ a point $(z_1^0,z_2^0)\in\SS^3\subset\CC^2$. Then $h(K_s)$ is a homotopy in $\mathbb{C}\backslash\{0\}$ from $h(K)$ to a point $h(z_1^0,z_2^0)$. The rotation number of $K$ is, by definition, the winding number of the loop $h(K)$. Since the winding number is a homotopy invariant in $\mathbb{C}\backslash\{0\}$ and a constant loop has vanishing winding number, we conclude that $\cR(K)=0$.
Conversely, suppose that $L=\cup_{i=1}^n L_i$ is a Legendrian link on $\SS^3$ (standard contact structure) with $\cR(L_i)=0$ for all $i$. We can safely assume that $L$ is real-analytic (up to a Legendrian isotopy)~\cite{rudolphtt}, so using~\cite[Lemma~2]{bode} we can define a real-analytic function $H_i:L_i\to\mathbb{C}\backslash\{0\}$ for each component of the link, so that a nonzero tangent vector of $L_i$ at any point on $L_i$ is given by
\begin{equation}\label{eq.tang}
\text{Re}(H_i)v_1-\text{Im}(H_i)v_2\,.
\end{equation}
Since $L_i$ has vanishing rotation number, the winding number of the loop $H_i(L_i)\subset\CC\backslash\{0\}$ is zero and hence $\text{Log}(H_i)$ is a well defined real-analytic function on $L_i$.
Now, the fact that $L_i$ is a real-analytic Legendrian knot and $\text{Log}(H_i)$ is real-analytic, allows us to apply directly the extension theorem by Burns and Stout~\cite{burns}. Therefore, we conclude that $\text{Log}(H_i)$ extends to a holomorphic function $h$ on a neighbourhood $U$ of $\SS^3\subset\CC^2$, i.e., $h|_{L_i}=\text{Log}(H_i)$ for all $i$.
Finally, we take the exponential of $h$, that is
$$\e^{h}:U\to\mathbb{C}\backslash\{0\}\,,$$
which is a holomorphic function that satisfies $\e^{h}|_{L_i}=H_i$. It then follows from Equation~\eqref{eq.tang} that the real part of
\begin{equation*}
\widetilde F:=\e^{h}(v_1+\I v_2)
\end{equation*}
is a vector field $X$ on $\SS^3$ with each $L_i$ as a periodic orbit. By construction, $X$ is non-vanishing and Legendrian. Moreover, since $h$ is a holomorphic function in $U$ (and hence a CR-function), Equation~\eqref{eq.div} shows that $\Div X=0$. This completes the proof of Theorem~\ref{P:m2}.
To prove Corollary~\ref{cor:rot}, we first observe (as discussed in Section~\ref{sec:bateman}) that the image of $L$ by the map $(\alpha,\beta)|_{t=0}:\RR^3\cup\{\infty\}\to \SS^3$, which is a diffeomorphism, is a link $L'\subset\SS^3$. Since any link on $\SS^3$ is isotopic to a Legendrian link (with respect to the standard contact structure) all of whose components have vanishing rotation number, see e.g.~\cite[Section~2.7]{etnyre}, we can safely assume (up to isotopy) that $L$ is a link in $\RR^3$ whose corresponding link $L'$ satisfies the aforementioned properties.
Now we can apply the previous construction with the components $L'_i$, $i\in I$, and for the other components of $L'$ we define a real-analytic function $H_i:L'_i\to\mathbb{C}\backslash\{0\}$, $i\not\in I$, so that a nonzero tangent vector is given by
\begin{equation*}
\text{Im}(H_i)v_1+\text{Re}(H_i)v_2\,.
\end{equation*}
Arguing as before we obtain a common holomorphic extension $h:U\to\CC^2$, $h|_{L'_i}=H_i$ for all $i$, and we can then define the Bateman field on $\RR^{3+1}$
\begin{equation*}
F=\e^{h(\alpha,\beta)}(\nabla\alpha\times\nabla\beta)\,,
\end{equation*}
which is non-vanishing by construction. We know from Section~\ref{sec:bateman} that, for each $t$, the pushforward by $(\alpha,\beta)|_t$ of the electric and magnetic parts of $F$ are precisely the real and imaginary parts of $\widetilde F$, up to a non-vanishing ($t$-dependent) proportionality factor. Since $\cup_{i\in I} L'$ is a subset of periodic orbits of $\text{Re}\,\widetilde F$, and the remaining components of $L'$ are a subset of periodic orbits of $\text{Im}\,\widetilde F$, we finally conclude that, for all $t$, the electric part of $F$ has a subset of periodic orbits isotopic to $\cup_{i\in I}L_i$, and the other components correspond to periodic orbits of the magnetic part. The corollary then follows.
\subsection{Legendrian Seifert foliations}\label{sec:seifert}
The Bateman solutions in Corollary~\ref{cor:rot} exhibit a prescribed set of knotted periodic orbits, but certainly not all electric or magnetic lines are closed. In light of Ra{\~n}ada's solution~\cite{R}, which corresponds to the Hopf fibration on $\SS^3$, it is natural to ask if such topological configurations of null electromagnetic fields can occur for other link types. To be precise, we are looking for an electric and/or magnetic field with the property that all the orbits of the corresponding Bateman-type Legendrian field on $\SS^3$ are closed.
A classical result of Epstein~\cite{epstein} shows that a vector field on $\SS^3$ all of whose orbits are closed forms a Seifert foliation. Every regular fiber of a Seifert foliation of $\SS^3$ is a torus link and the two exceptional fibers always form a Hopf link. The family of Seifert foliations of $\SS^3$ is completely characterized by two integers $p$ and $q$, $q\geq1$, with $\gcd(p,q)=1$. The leaf of the foliation $S_{p,q}$ through a point $(z_1,z_2)\in \SS^3$ is given as the parametric curve
\begin{equation*}
(z_1\e^{\I p\varphi},z_2\e^{\I q\varphi})\,,
\end{equation*}
where $\varphi\in[0,2\pi)$. The leaf is a $(p,q)$-torus knot unless $z_1=0$ or $z_2=0$, in which case it is an unknot.
Any two such foliations $S_{p,q}$ and $S_{p',q'}$ with $(p,q)\neq(p',q')$ and $\gcd(p,q)=\gcd(p',q')=1$ are not equivalent, while every Seifert foliation of $\SS^3$ is equivalent to $S_{p,q}$ for some choice of $(p,q)$. We recall that two foliations are said to be equivalent if there is an orientation-preserving diffeomorphism mapping one to the other.
We infer from Theorem~\ref{P:m1} that a necessary condition for a Seifert foliation to be of Bateman-type is that all its leaves must be Legendrian with respect to the standard contact structure. See~\cite{liebermann,pang} for a general study of~\textit{Legendrian foliations}. Here we prove Proposition~\ref{P:seifert}, i.e., that every positive Seifert foliation $S_{p,q}$, $p>0$, $q>0$, is equivalent to one that is Legendrian with respect to the standard contact structure.
\begin{proof}[Proof of Proposition~\ref{P:seifert}]
A vector field $X_{p,q}$ tangent to the Seifert foliation $S_{p,q}$ of $\SS^3$ is given by
\begin{equation*}
X_{p,q}=-py_1\partial_{x_1}+px_1\partial_{y_1}-qy_2\partial_{x_2}+qx_2\partial_{y_2}\,,
\end{equation*}
where we are using coordinates $z_i=x_i+\I y_i$, $i=1,2$. Obviously, $X_{p,q}$ is divergence-free. A straightforward computation shows that $X_{p,q}$ is in the kernel of the $1$-form
\begin{align}
\alpha_{p,q}:=&\left(-\frac{(p (x_1^2 + y_1^2) + q (x_2^2 + y_2^2)) (x_1 x_2 - y_1 y_2) x_1}{x_1^2 + y_1^2} - q \frac{(y_1 x_2 + x_1 y_2) y_1}{x_1^2 + y_1^2}\right)\dd x_1\nonumber\\
&+\left(-\frac{(p (x_1^2 + y_1^2) + q (x_2^2 + y_2^2)) (x_1 x_2 - y_1 y_2) y_1}{x_1^2 + y_1^2} + q \frac{(y_1 x_2 + x_1 y_2) x_1}{x_1^2 + y_1^2}\right)\dd y_1\nonumber\\
&+\left(\frac{(p (x_1^2 + y_1^2) + q (x_2^2 + y_2^2)) (x_1 x_2 - y_1 y_2) x_2}{x_2^2 + y_2^2} + p \frac{(y_1 x_2 + x_1 y_2) y_2}{x_2^2 + y_2^2}\right)\dd x_2\nonumber\\
&+\left(\frac{(p (x_1^2 + y_1^2) + q (x_2^2 + y_2^2)) (x_1 x_2 - y_1 y_2) y_2}{x_2^2 + y_2^2} - p \frac{(y_1 x_2 + x_1 y_2) x_2}{x_2^2 + y_2^2}\right)\dd y_2\,.
\end{align}
This $1$-form is smooth everywhere (actually, real-analytic). This can be easily seen using Hopf coordinates $(z_1,z_2)=(\cos s\exp(\I\phi_1),\sin s \exp(\I\phi_2))$, $s\in[0,\pi/2]$, $\phi_{1,2}\in[0,2\pi)$, where $\alpha_{p,q}$ takes the simpler form
\[
\alpha_{p,q}=(p\cos^2 s+q\sin^2 s)\cos(\phi_1+\phi_2)ds+\sin(\phi_1+\phi_2)\sin s\cos s (qd\phi_1-pd\phi_2)\,,
\]
and $X_{p,q}$ reads as
\[
X_{p,q}=p\partial_{\phi_1}+q\partial_{\phi_2}\,.
\]
We claim that the 1-form $\alpha_{p,q}$ is contact (and hence its kernel defines a contact structure $\xi_{p,q}$) if $p>0$, $q>0$. Indeed, an easy computation in Hopf coordinates yields
\begin{align*}
\alpha_{p,q}\wedge d\alpha_{p,q}=(p+q)(p\cos^2s+q\sin^2s)\Big(-\sin s\cos s ds\wedge d\phi_1\wedge d\phi_2\Big)\\
=(p+q)(p\cos^2s+q\sin^2s)\mu_0\,,
\end{align*}
where $\mu_0$ is the standard volume-form on $\SS^3$ written in Hopf coordinates. It is then clear that $\alpha_{p,q}$ is a (real-analytic) contact form for any choice of $p>0$ and $q>0$, even if these numbers take non-integer values.
In view of the expression of $\alpha_{p,q}$ it is easy to construct an homotopy of contact forms that connects $\alpha_{p,q}$ with $\alpha_{1,1}$:
\[
\alpha(t):=\alpha_{p(t),q(t)}
\]
with $p(t)=t+p(1-t)$ and $q(t)=t+q(1-t)$, $t\in[0,1]$. Obviously $\alpha(0)=\alpha_{p,q}$ and $\alpha(1)=\alpha_{1,1}$. Moser's path method~\cite{Geiges} then implies that there is a (contact) isotopy $\Phi_t$ of diffeomorphisms that transforms the corresponding contact structures. In fact, since
\begin{equation*}
\alpha_{1,1}=-x_2\dd x_1+y_2\dd y_1+x_1 \dd x_2-y_1 \dd y_2\,,
\end{equation*}
is obtained from the standard contact form on $\SS^3$ by the rotation $(x_1,y_1,x_2,y_2)\mapsto(x_1,-x_2,y_1,y_2)$, we infer that all the contact structures $\xi_{p,q}$, $p>0$, $q>0$, are equivalent to the standard contact structure on $\SS^3$. Since the homotopy of contact forms $\alpha(t)$ is real analytic in $t$ and on $\SS^3$, the proof of Moser's theorem shows that $\Phi_t$ is a real-analytic isotopy.
Denoting by $\Phi_{p,q}$ the real-analytic diffeomorphism of $\SS^3$ such that $\xi_{p,q}$ is the pullback of the standard contact structure, we infer that the pushforward $(\Phi_{p,q})_*(X_{p,q})$ is a real-analytic Legendrian vector field with respect to the standard contact structure. Moreover, since $X_{p,q}$ is divergence-free, i.e., it preserves the standard volume form,
\[
L_{X_{p,q}}\mu_0=0\,,
\]
we conclude that $(\Phi_{p,q})_*(X_{p,q})$ preserves the volume form $(\Phi_{p,q}^{-1})^*\mu_0$. Defining the positive real-analytic function $F_{p,q}$ on $\SS^3$ as
\[
(\Phi^{-1}_{p,q})^*\mu_0=:F_{p,q}\mu_0\,,
\]
we obtain that the vector field $X:=F_{p,q}(\Phi_{p,q})_*(X_{p,q})$ is real-analytic, divergence-free and Legendrian, with respect to the standard contact structure. The proposition then follows.
\end{proof}
\begin{remark}
In principle, Moser's path method also provides us with instructions how to find the diffeomorphism $\Phi_{p,q}$. We could then check the divergence of the Legendrian orthogonal field $JX$. If $\Div(JX)=0$, then $(X,JX)$ is a pair of Legendrian fields of Bateman type and hence the topological structure is stable for all~$t$. In practice however, the differential equation that needs to be solved following Moser's approach is hard to analyze, so that at this stage the question whether Seifert foliations other than the Hopf field arise as stable structures in electromagnetic fields remains unsolved.
\end{remark}
\section{New Bateman variables and torus knots}\label{sec:torus}
In this section we construct a pair of Legendrian vector fields of Bateman type on $\SS^3$ such that the electric part realizes all positive torus knots in its set
of periodic orbits. This construction is achieved by introducing a new couple of Bateman variables, i.e., new solutions to the Equation~\eqref{eq:1}, which is presented in Section~\ref{SS.rever}.
\subsection{Antiholomorphic functions and time reversal}\label{SS.rever}
Let $\alpha:\mathbb{R}^{3+1}$ be a complex-valued function in the three spatial variables $x$, $y$ and $z$, and the temporal variable $t$. Then we denote by $\tilde{\alpha}:\mathbb{R}^{3+1}\to\mathbb{C}$ the function that is obtained from $\alpha$ by taking its complex conjugate and reversing time, i.e., $t$ is substituted by $-t$:
\[
\tilde\alpha(x,y,z,t):=\overline{\alpha(x,y,z,-t)}\,.
\]
The following lemma shows that this elementary (albeit nontrivial) transformation preserves the Bateman property.
\begin{lemma}\label{L.newBat}
Let $\alpha$ and $\beta$ be Bateman variables. Then $\tilde{\alpha}$ and $\tilde{\beta}$ as defined above are also Bateman variables.
\end{lemma}
\begin{proof}
By definition of Bateman variables, the spacetime functions $\alpha$ and $\beta$ satisfy Equation~\eqref{eq:1}. We claim that the couple $(\tilde\alpha,\tilde\beta)$ also satisfies Equation~\eqref{eq:1}. Indeed
\begin{align}
\nabla\tilde{\alpha}\times\nabla\tilde{\beta}&=\left(\overline{\nabla\alpha\times\nabla\beta}\right)|_{t\to-t}\nonumber\\
&=-\I \overline{(\partial_t\alpha\nabla\beta-\partial_t\beta\nabla\alpha)}|_{t\to-t}\nonumber\\
&=-\I (\partial_t\overline{\alpha}\nabla\overline{\beta}-\partial_t\overline{\beta}\nabla\overline{\alpha})|_{t\to-t}\nonumber\\
&=\I (\partial_t\tilde{\alpha}\nabla\tilde{\beta}-\partial_t\tilde{\beta}\nabla\tilde{\alpha})\,,
\end{align}
which completes the proof of the lemma.
\end{proof}
Next, we use this lemma to construct new null solutions to Maxwell's equations that are not of Hopf type. We recall that a function $h:U\subset\CC^2\to \CC$ is antiholomorphic if its complex conjugate $\overline{h}$ is holomorphic.
\begin{proposition}\label{P:anti}
Let $(\alpha,\beta)$ be Bateman variables such that the image of $(\overline{\alpha},\overline{\beta})$ is contained (for all $t$) in the image of $(\alpha,\beta)|_{t=0}$. Let $h:U\to\mathbb{C}$ be an antiholomorphic function on some neighbourhood $U$ of the image of $(\alpha,\beta)|_{t=0}$ in $\mathbb{C}^2$. Then $$E_0:=\text{Re}(h(\alpha,\beta)\nabla\alpha\times\nabla\beta)|_{t=0}\,, \qquad B_0:=-\text{Im}(h(\alpha,\beta)\nabla\alpha\times\nabla\beta)|_{t=0}$$
are the initial data of an electromagnetic field that is null for all time.
\end{proposition}
\begin{proof}
We first calculate
\begin{align}\label{eq.other}
E_0+\I B_0&=\overline{h(\alpha,\beta)\nabla\alpha\times\nabla\beta}|_{t=0}\nonumber\\
&=\overline{h}(\overline{\alpha},\overline{\beta})\nabla\overline{\alpha}\times\nabla\overline{\beta}|_{t=0}\nonumber\\
&=\overline{h}(\tilde{\alpha},\tilde{\beta})\nabla\tilde{\alpha}\times\nabla\tilde{\beta}|_{t=0}\,,
\end{align}
where $\tilde\alpha$ and $\tilde\beta$ are defined as before. Since $h$ is antiholomorphic on $U$, its conjugate $\overline{h}$ is holomorphic on a neighbourhood of the image of $(\overline{\alpha},\overline{\beta})$, which is the same as the image of $(\tilde{\alpha},\tilde{\beta})$. Since $\tilde{\alpha}$ and $\tilde{\beta}$ are Bateman variables, cf. Lemma~\ref{L.newBat}, the initial datum $E_0+\I B_0$ is equal to the corresponding Bateman field at $t=0$. The proposition then follows from Bateman construction and the uniqueness of solutions.
\end{proof}
\begin{remark}
It is important to emphasize that Proposition~\ref{P:anti} does not contradict Bateman's construction presented in Section~\ref{sec:bateman} (where we required holomorphicity of $h$). It is true that a pair of time dependent vector fields on $\mathbb{R}^3$ describes a Bateman field associated with the Hopf fibration if and only if the corresponding function $h$ on $\SS^3$ satisfies the tangential Cauchy-Riemann equations on $\SS^3$. However, in Proposition~\ref{P:anti} we construct null solutions to Maxwell's equations which are of Bateman type not for $(\alpha,\beta)$ from Equation~\eqref{eq:stereo3}, but for the new Bateman variables $(\tilde\alpha,\tilde\beta)$ (which are not of Hopf type). Accordingly, as shown in Equation~\eqref{eq.other}, it is $\overline{h}$ the function that must be holomorphic, and hence $h$ is antiholomorphic.
\end{remark}
We finally observe that the Bateman pair $(\alpha,\beta)$ introduced in Equation~\eqref{eq:stereo3} (Bateman solution of Hopf type) satisfies the assumption in Proposition~\ref{P:anti} because the image of $(\alpha,\beta)|_{t=t_*}$ is $\SS^3$ for all $t_*\in\mathbb{R}$. It then follows that pairs of Legendrian vector fields on $\SS^3$ whose components with respect to the Hopf basis $\{v_1,v_2\}$ are the real and imaginary part of an antiholomorphic function also correspond to Bateman fields.
\subsection{Torus knots: proof of Proposition~\ref{P:torus}}\label{sec:dense}
In this section we prove Proposition~\ref{P:torus} using the results obtained in the previous section. To this end, we consider the antiholomorphic function $h(z_1,z_2)=\overline{z_1}\overline{z_2}$. The Legendrian vector field $X$ on $\SS^3$ given by $\text{Re}(h(z_1,z_2)(v_1+\I v_2))$ can be explicitly written as
\begin{equation}
X=y_1 (x_2^2+y_2^2)\partial_{x_1}-x_1(x_2^2+y_2^2)\partial_{y_1}-y_2(x_1^2+y_1^2)\partial_{x_2}+x_2(x_1^2+y_1^2)\partial_{y_2}\,,
\end{equation}
where $z_i=x_i+\I y_i$ as usual. It is easy to check that the functions $|z_1|^2$ and $|z_2|^2$ are first integrals of the vector field $X$. Using this observation, it is elementary to compute the integral curve of $X$ through a point $(z_1,z_2)\in\SS^3$, which is given by
\begin{equation}
(z_1\e^{-\I |z_2|^2\tau},z_2\e^{\I |z_1|^2\tau})\,,
\end{equation}
for $\tau\in\RR$.
The field lines of $X$ are tangent to the tori of constant $|z_1|$ (or, equivalently, constant $|z_2|$) in $\SS^3$. In particular, the Hopf link defined by $\{z_1z_2=0\}$ is a set of two periodic orbits of $X$. The slope of the field lines is given by $-|z_2|^2/|z_1|^2$, so that the field lines are closed if and only if $|z_2|^2/|z_1|^2\in\mathbb{Q}\cup\{\infty\}$. In particular, for $|z_2|^2/|z_1|^2=p/q$ with $p,q\in\mathbb{N}$, all field lines on the corresponding torus form the torus knot $T_{p,-q}$.
Recall that for the Bateman variables $(\alpha,\beta)$ as in Equation~\eqref{eq:stereo3} we denote the inverse of $(\alpha,\beta)|_{t=0}$ by $\phi_0$. The pushforward $E_0$ of $X$ by $\phi_0$ is up to a real factor the real part of $h(\alpha,\beta)(\nabla\alpha\times\nabla\beta)|_{t=0}$. Then, by Proposition~\ref{P:anti}, $E_0$ and $B_0:=-\text{Im}(h(\alpha,\beta)(\nabla\alpha\times\nabla\beta)|_{t=0})$ form the initial datum of a Bateman field with Bateman variables $(\tilde{\alpha},\tilde{\beta})$. In particular, the topological structure of nested tori and the foliation by torus knots on a dense subset of these tori is preserved for all time. Note that $\phi_0$ is orientation reversing, since it is the composition of a stereographic projection and a mirror reflection. Therefore, the negative torus knot $T_{p,-q}$ in $\SS^3$ is transformed into its mirror image $T_{p,q}$ in $\mathbb{R}^3$. This completes the proof of the Proposition.
\section{Final remarks on integrable Bateman fields}\label{sec:int}
In this final section we explore the integrability properties (i.e., the existence of first integrals) of Bateman solutions of Hopf type. Let us first recall that the Bateman fields in~\cite{kbbpsi}, which contain torus knots, correspond to the holomorphic function $h=pqz_1^{p-1}z_2^{q-1}$, and can be equivalently written as
\begin{equation}
F=\nabla f(\alpha,\beta)\times\nabla g(\alpha,\beta)\,,
\end{equation}
with $(\alpha,\beta)$ as in Equation~\eqref{eq:stereo3} and $f=z_1^p$ and $g=z_2^q$.
Besides the fact that these fields have periodic orbits in the shape of a $(p,q)$-torus knot, these fields possess a surprising integrability property, that is, $E=\text{Re}(F)$ is tangent to the level sets of $\text{Im}(fg)$ and $B=\text{Im}(F)$ is tangent to the level sets of $\text{Re}(fg)$ (both functions are first integrals of the corresponding fields). This can be proved by a direct calculation. In what follows we want to explain this integrability property in a more general way that relates the electromagnetic fields to the concept of totally tangential $\mathbb{C}$-links.
\subsection{Legendrian fields with prescribed first integrals}
First, we introduce a method (of interest in itself) to construct Legendrian vector fields on $\SS^3$ (with respect to the standard contact structure) with a prescribed first integral.
\begin{lemma}\label{L.fi}
Let $G_{\mathbb{R}}:\SS^3\to\mathbb{R}$ be a smooth function and define the complex-valued function
\begin{equation}\label{eq:htang}
h(z_1,z_2)=\left(\frac{\partial G_{\mathbb{R}}}{\partial x_2}-\I \frac{\partial G_{\mathbb{R}}}{\partial y_2}\right)(x_1-\I y_1)-\left(\frac{\partial G_{\mathbb{R}}}{\partial x_1}-\I \frac{\partial G_{\mathbb{R}}}{\partial y_1}\right)(x_2-\I y_2)\,.
\end{equation}
Then $G_{\mathbb{R}}$ is a first integral of the Legendrian field $\text{Im}(h(v_1+\I v_2))$.
Furthermore, if $G_{\mathbb{R}}$ extends to a pluriharmonic function on some neighbourhood $U$ of $\SS^3$, i.e., there is a holomorphic function $G:U\to\mathbb{C}$ with $\text{Re}(G)|_{\SS^3}=G_{\mathbb{R}}$, then $\text{Im}(G)|_{\SS^3}$ is a first integral of the Legendrian field $\text{Re}(h(v_1+\I v_2))$.
\end{lemma}
\begin{proof}
Consider the complex structure $J$ on $\mathbb{C}^2\cong\mathbb{R}^4$, which on each real tangent space $T_p\mathbb{R}^4=\mathbb{R}^4$ maps $(a_1,a_2,a_3,a_4)$ to $J(a_1,a_2,a_3,a_4)=(-a_2,a_1,-a_4,a_3)$.
A direct calculation shows that
\begin{align}
\text{Re}(h(z_1,z_2))&=\nabla G_{\mathbb{R}}\cdot v_1=J\nabla G_{\mathbb{R}}\cdot v_2\,,\label{eq.h1}\\
\text{Im}(h(z_1,z_2))&=-\nabla G_{\mathbb{R}}\cdot v_2=J\nabla G_{\mathbb{R}}\cdot v_1\,,\label{eq.h2}
\end{align}
where $\nabla$ denotes the gradient $(\partial_{x_1},\partial_{y_1},\partial_{x_2},\partial_{y_2})$, and $\{v_1,v_2\}$ are the Hopf fields introduced in Section~\ref{sec:bateman}. It follows that
\begin{equation}
\text{Im}(h(v_1+\I v_2))=(J\nabla G_{\mathbb{R}}\cdot v_1)v_1+(J\nabla G_{\mathbb{R}}\cdot v_2)v_2\,,
\end{equation}
or equivalently, $\text{Im}(h(v_1+\I v_2))$ is the projection of $J\nabla G_{\mathbb{R}}$ into the contact plane at each point of $\SS^3$, spanned by $v_1$ and $v_2$.
Now we change the basis of each tangent space from ${\partial_{x_1}, \partial_{y_1}, \partial_{x_2}, \partial_{y_2}}$ to
\begin{equation}
\{v_1,v_2,v_3,v_4\}:=\{v_1,v_2,x_1\partial_{x_1}+y_1\partial_{y_1}+x_2\partial_{x_2}+y_2\partial_{y_2},-y_1\partial_{x_1}+x_1\partial_{y_1}-y_2\partial_{x_2}+
x_2\partial_{y_2}\}\,.
\end{equation}
Note that this defines an orthonormal basis at each point of $\SS^3$, and the change of basis preserves orthogonality. Further note that $J(v_1)=v_2$ and $J(v_3)=v_4$, so that in this new basis $J$ still takes the form $J(a_1,a_2,a_3,a_4)=(-a_2,a_1,-a_4,a_3)$. So if $\nabla G_{\mathbb{R}}=(a_1,a_2,a_3,a_4)$ in this basis, then $J(\nabla G_{\mathbb{R}})=(-a_2,a_1,-a_4,a_3)$ and $\text{Im}(h(v_1+\I v_2))=(-a_2,a_1,0,0)$, which is clearly orthogonal to $\nabla G_{\mathbb{R}}$. Therefore, $G_{\mathbb R}$ is a first integral of the vector field $\text{Im}(h(v_1+\I v_2))$.
If $G_\mathbb{R}$ is the real part of a holomorphic function $G$, Equation~\eqref{eq:htang} becomes
\begin{equation}\label{eq.fig}
h(z_1,z_2)=2\frac{\partial G}{\partial z_2}\overline{z_1}-2\frac{\partial G}{\partial z_1}\overline{z_2}
\end{equation}
and
\begin{equation}
J(\nabla G_{\mathbb{R}})=J(\nabla \text{Re}(G))=\nabla\text{Im}(G)\,.
\end{equation}
Therefore, we obtain
\begin{align}
\text{Re}(h(v_1+\I v_2))\cdot\nabla\text{Im}(G)&=J(\text{Re}(h(v_1+\I v_2)))\cdot J(\nabla\text{Im}(G))\nonumber\\
&=-\text{Im}(h(v_1+\I v_2))\cdot \nabla(G_{\RR})=0\,,
\end{align}
thus implying that $\text{Im}(G)$ is a first integral of the vector field $\text{Re}(h(v_1+\I v_2))$.
\end{proof}
Next we use this lemma to provide a criterion for the existence of invariant surfaces for Legendrian fields on $\SS^3$. We recall that a level set of a function is regular if the gradient of the function does not vanish at any point of the level set.
\begin{proposition}\label{P.arg}
A Legendrian field $X=\text{Im}(\Th(v_1+\I v_2))$ with $\Th:\SS^3\to\mathbb{C}$, is tangent to the regular level set $M=G_{\mathbb{R}}^{-1}(c)$, $c\in\mathbb{R}$, of a real function $G_{\mathbb{R}}:\SS^3\to\mathbb{R}$ if and only if
\begin{equation}
\arg(\Th(z_1,z_2))=\arg(h(z_1,z_2))
\end{equation}
for all $(z_1,z_2)\in M$ with $h(z_1,z_2)\neq 0$, where $h$ is given by Equation~\eqref{eq:htang}.
\end{proposition}
\begin{proof}
It follows from Equations~\eqref{eq.h1} and~\eqref{eq.h2} that $h(z_1,z_2)=0$ if and only if $\nabla G_{\mathbb{R}}$ is orthogonal to the contact plane at the point $(z_1,z_2)\in\SS^3$. In particular, any Legendrian vector field (with respect to the standard contact form) is tangent to $M$ at a point $(z_1,z_2)$ if $h(z_1,z_2)=0$. Next we analyze the points $(z_1,z_2)\in M$ such that $h(z_1,z_2)\neq 0$.
We have already seen, cf. Lemma~\ref{L.fi}, that $\text{Im}(h(v_1+\I v_2))$ is tangent to $M$. It then follows that if $\arg(\Th(z_1,z_2))=\arg(h(z_1,z_2))$ for some $(z_1,z_2)\in M$, then $X$ is tangent to $M$ at $(z_1,z_2)$. Conversely, if $h(z_1,z_2)\neq 0$, the intersection of the contact plane at $(z_1,z_2)$ and $T_{(z_1,z_2)}M$ is 1-dimensional (because the level set is regular), so that any Legendrian tangent vector to $M$ has to be a real multiple of $\text{Im}(h(v_1+\I v_2))$ at $(z_1,z_2)$, and hence $\arg(\Th(z_1,z_2))=\arg(h(z_1,z_2))$.
\end{proof}
In particular, the following corollary is straightforward from Bateman construction and Proposition~\ref{P.arg}. It provides a criterion to construct null electromagnetic fields with prescribed invariant surfaces (e.g., invariant tori).
\begin{corollary}
Let $G_{\mathbb{R}}:\SS^3\to\mathbb{R}$ be a smooth function. Then there is a Bateman electromagnetic field of Hopf type, whose magnetic part at $t=0$ (when projected onto $\SS^3$) is tangent to the regular level set $M:=G_{\mathbb{R}}^{-1}(c)$, $c\in\RR$, if and only if there is a holomorphic function $\Th:U\to \mathbb{C}$ on some neighbourhood $U$ of $\SS^3$ with
\begin{equation}\label{eq:th}
\arg(\Th(z_1,z_2))=\arg(h(z_1,z_2))
\end{equation}
for all $(z_1,z_2)\in M$ with $h(z_1,z_2)\neq 0$, where $h$ is given by Equation~\eqref{eq:htang}.
\end{corollary}
\subsection{Totally tangential $\CC$-links and Bateman}
Let us start with the following definition:
\begin{definition}[Rudolph \cite{rudolph}]
A link $L$ in $\SS^3$ is called a totally tangential $\mathbb{C}$-link if there exists a holomorphic function $G:\mathbb{C}^2\to\mathbb{C}$ such that
\begin{itemize}
\item $G^{-1}(0)\cap\SS^3=L$.
\item $G^{-1}(0)\cap \mathring{D}^4=\emptyset$, where $\mathring{D}^4$ denotes the open 4-ball bounded by $\SS^3$ in $\CC^2$.
\item $L$ is a non-degenerate critical manifold of index 1 of $\rho|_{\text{Reg}(G)}$, the restriction of $\rho(z_1,z_2):=|z_1|^2+|z_2|^2$ to the set of regular points of $G$.
\end{itemize}
\end{definition}
\begin{remark}
It is known that a link in $\SS^3$ is a totally tangential $\mathbb{C}$-link if and only if it is real-analytic and Legendrian~\cite{rudolphtt, rudolphtt2, rudolph}.
\end{remark}
In~\cite{bode} it was suggested that there could be a relation between the holomorphic function $G$ that defines $L$ as an intersection and the holomorphic function $h$, which defines the electromagnetic field containing $L$ (via Bateman construction). Next we shall show that in the case of torus knots~\cite{kbbpsi} we can make this relation precise, in a way that incorporates the integrability of the electromagnetic fields.
We first observe that the polynomial $G$ that defines the torus knot $T_{p,-q}$ as a totally tangential intersection of $G^{-1}(0)$ and $\SS^3$ is given by $$G(z_1,z_2)=\rho z_1^p z_2^q-1$$
for some appropriately chosen constant $\rho>0$ (see~\cite{rudolphtt}). For the class of electromagnetic solutions found in~\cite{kbbpsi}, the following result establishes a connection between the function $G$ and the first integrals of the fields.
\begin{proposition}\label{P.Gh}
The Legendrian field on $\SS^3$ that corresponds to the electric field $E$ in~\cite{kbbpsi} (containing the torus knot $T_{p,q}$ in $\mathbb{R}^3$, cf. Remark~\ref{rem}) is tangent to the level sets of $\text{Im}(G)$, and the Legendrian field corresponding to the associated magnetic field $B$ is tangent to the level sets of $\text{Re}(G)$.
\end{proposition}
\begin{proof}
Indeed, the Bateman fields constructed in~\cite{kbbpsi} use the complex polynomial
$$\tilde h(z_1,z_2)=pqz_1^{p-1}z_2^{q-1}\,,$$
while the $(p,q)$-torus knot arises as a totally tangential $\mathbb{C}$-link for the function $G$ introduced before. In order to construct a pair of Legendrian fields on $\SS^3$ for which $\text{Re}(G)$ and $\text{Im}(G)$ are first integrals, we use Lemma~\ref{L.fi} and define $h$ as in Equation~\eqref{eq.fig}, resulting in
\begin{equation}
h(z_1,z_2)=2\rho q z_1^p z_2^{q-1}\overline{z_1}-2\rho p z_1^{p-1}z_2^q\overline{z_2}=pq z_1^{p-1}z_2^{q-1}\Big(2\rho\frac{1}{p}|z_1|^2-2\rho\frac{1}{q}|z_2|^2\Big)\,.
\end{equation}
Ignoring the real factor $2\rho\frac{1}{p}|z_1|^2-2\rho\frac{1}{q}|z_2|^2$ we are left with the same holomorphic function $\tilde h$ as in~\cite{kbbpsi}. Accordingly, up to a real factor, the electric and magnetic parts of the Bateman field~\cite{kbbpsi}, when projected onto $\SS^3$, are tangent to the Legendrian fields obtained in Lemma~\ref{L.fi} using the function $h$. We then conclude that the magnetic and electric lines are tangent to the level sets of the real and imaginary part of $G$, respectively, as we wanted to prove.
\end{proof}
\begin{remark}
There is a topological obstruction regarding the link types that can arise as periodic orbits of non-vanishing vector fields with analytic first integrals. This class of links is called zero-entropy links and has been classified by Etnyre and Ghrist, following previous work by Wada~\cite{wada,EG}. Therefore, since $G$ is a holomorphic function, if the construction above generalizes to other link types, it must produce Bateman fields with zeros in the case of links that do not belong to the family of zero-entropy links.
\end{remark}
To conclude this section we want to emphasize that it would be interesting to see if the construction in Proposition~\ref{P.Gh} generalizes to other holomorphic functions $G$, resulting in other Bateman type solutions exhibiting first integrals. However, at this stage it remains difficult to determine the existence of a holomorphic function $\Th$ satisfying Equation~\eqref{eq:th}. In fact, the torus knots above are the only examples where the function $G$ is known explicitly (although it exists for every link type). Likewise, we know that there exist Bateman fields that realize electric and magnetic lines of any link type, but explicit expressions have only been found for torus links~\cite{kbbpsi}.
\section*{Acknowledgements}
The authors are grateful to R. Casals and A. del Pino for useful comments concerning Legendrian fields. B.B. is supported by the European Union's Horizon 2020 research and innovation programme through the Marie Sklodowska-Curie grant agreement 101023017.
This work is supported by the grants CEX2019-000904-S and PID2019-106715GB GB-C21 funded by MCIN/AEI/10.13039/501100011033.
\bibliographystyle{amsplain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,942 |
\section{#1}}
\renewcommand{\thesection.\arabic{equation}}}{\thesection.\arabic{equation}}
\def\titlepage{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
\else \newpage \fi \thispagestyle{empty}\c@page\z@
\def\arabic{footnote}{\fnsymbol{footnote}} }
\def\endtitlepage{\if@restonecol\twocolumn \else \fi
\def\arabic{footnote}{\arabic{footnote}}
\setcounter{footnote}{0}}
\relax
\hybrid
\parskip=0.4em
\makeatletter
\newdimen\normalarrayskip
\newdimen\minarrayskip
\normalarrayskip\baselineskip
\minarrayskip\jot
\newif\ifold \oldtrue \def\oldfalse{\oldfalse}
\def\arraymode{\ifold\relax\else\displaystyle\fi
\def\eqnumphantom{\phantom{(\thesection.\arabic{equation}})}}
\def\@arrayskip{\ifold\baselineskip\z@\lineskip\z@
\else
\baselineskip\minarrayskip\lineskip1\baselineskip\fi}
\def\@arrayclassz{\ifcase \@lastchclass \@acolampacol \or
\@ampacol \or \or \or \@addamp \or
\@acolampacol \or \@firstampfalse \@acol \fi
\edef\@preamble{\@preamble
\ifcase \@chnum
\hfil$\relax\arraymode\@sharp$\hfil
\or $\relax\arraymode\@sharp$\hfil
\or \hfil$\relax\arraymode\@sharp$\fi}}
\def\@array[#1]#2{\setbox\@arstrutbox=\hbox{\vrule
height\arraystretch \ht\strutbox
depth\arraystretch \dp\strutbox
width\z@}\@mkpream{#2}\edef\@preamble{\halign \noexpand\@halignto
\bgroup \tabskip\z@ \@arstrut \@preamble \tabskip\z@ \cr}%
\let\@startpbox\@@startpbox \let\@endpbox\@@endpbox
\if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi
\bgroup \let\par\relax
\let\@sharp##\let\protect\relax
\@arrayskip\@preamble}
\def\eqnarray{\stepcounter{equation}%
\let\@currentlabel=\thesection.\arabic{equation}}
\global\@eqnswtrue
\global\@eqcnt\z@
\tabskip\@centering
\let\\=\@eqncr
$$%
\halign to \displaywidth \bgroup
\eqnumphantom \@eqnsel
\hskip\@centering
$\displaystyle \tabskip\z@ {##}$%
&\global\@eqcnt\@ne \hskip 2\arraycolsep
$ \displaystyle \arraymode{##}$\hfil
&\global\@eqcnt\tw@ \hskip 2\arraycolsep
$\displaystyle\tabskip\z@{##}$\hfil
\tabskip\@centering
&{##}\tabskip\z@\cr}
\makeatother
\newtheorem{te}{Theorem}[section
\newtheorem{de}{Definition}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{ex}{Example}[section]
\newtheorem{rem}{Remark}[section]
\newcommand{\begin{equation}}[1]{\begin{equation}\label{#1}}
\newcommand\end{equation}{\end{equation}}
\newcommand\bqa{\begin{eqnarray}}
\newcommand\eqa{\end{eqnarray}}
\def\begin{eqnarray}\new\begin{array}{cc}{\begin{eqnarray}\oldfalse\begin{array}{cc}}
\def\end{array}\end{eqnarray}{\end{array}\end{eqnarray}}
\def\nonumber{\nonumber}
\defg{g}
\def{\cal U}{{\cal U}}
\def\begin{equation}{\begin{equation}}
\def\end{equation}{\end{equation}}
\def\bse{\begin{subequations}}
\def\end{subequations}{\end{subequations}}
\def\begin{pmatrix}{\begin{pmatrix}}
\def\end{pmatrix}{\end{pmatrix}}
\def\be\label{\begin{eqnarray}\new\begin{array}{cc}\label}
\def\hbar{\hbar}
\def\imath{\imath}
\defQ\!\!\!\! Q{Q\!\!\!\! Q}
\def{\bf M}{{\bf M}}
\newcommand\cod{\operatorname{codim}}
\newcommand\im{\operatorname{im}}
\newcommand\id{\operatorname{id}}
\newcommand\coim{\operatorname{coim}}
\newcommand\rk{\operatorname{rank}}
\newcommand\ann{\operatorname{ann}}
\newcommand{\mathbb{R}}{\mathbb{R}}
\newcommand{\mathbb{C}}{\mathbb{C}}
\newcommand{\mathbb{Q}}{\mathbb{Q}}
\newcommand{\mathbb{F}}{\mathbb{F}}
\newcommand{\overline{\mathbb{C}}}{\overline{\mathbb{C}}}
\newcommand{\mathbb{Z}}{\mathbb{Z}}
\newcommand{{\bf g}}{{\bf g}}
\def\square{\hfill{\vrule height6pt width6pt
depth1pt} \break \vspace{.01cm}}
\def\ws{\hfill{$\square$}}
\def\partial{\partial}
\def\stack#1#2{\raise0.7pt\hbox{$\mathrel{\mathop{#2}\limits^{#1}}$}}
\def\triangleright{\triangleright}
\def\triangleleft{\triangleleft}
\def\sem{\mathsurround=0pt \raise1pt
\hbox{$\scriptscriptstyle>\!\!$}\:\!\!\triangleleft}
\def\mes{\mathsurround=0pt \triangleright\!\:\!\raise0.8pt
\hbox{$\scriptscriptstyle\!\!<$}\,}
\def\]{\mathsurround=0pt ]\raise-2pt\hbox{$_\ast$}}
\def{\bfit\alpha}{{\bfit\alpha}}
\def{\bfit\beta}{{\bfit\beta}}
\def{\bfit\gamma}{{\bfit\gamma}}
\def\bnu{{\bfit\nu}}
\def{\bfit\mu}{{\bfit\mu}}
\def{\bfit\omega}{{\bfit\omega}}
\def{\bfit\phi}{{\bfit\phi}}
\def{\bfit\lambda}{{\bfit\lambda}}
\def{\bfit\rho}{{\bfit\rho}}
\def\alpha{\alpha}
\def\lambda{\lambda}
\def\varepsilon{\varepsilon}
\def\epsilon{\epsilon}
\def\<{\langle}
\def\>{\rangle}
\def\overline{\overline}
\def\widetilde{\widetilde}
\def\widehat{\widehat}
\def\varkappa{\varkappa}
\def\mathfrak{\mathfrak}
\def{\cal A}{{\cal A}}
\def{\cal B}{{\cal B}}
\def{\cal C}{{\cal C}}
\def{\cal D}{{\cal D}}
\def{\cal H}{{\cal H}}
\def{\cal M}{{\cal M}}
\def{\cal L}{{\cal L}}
\def{\cal O}{{\cal O}}
\def{\cal R}{{\cal R}}
\def{\cal S}{{\cal S}}
\def\mathcal{H}{\mathcal{H}}
\def{\scriptscriptstyle N}{{\scriptscriptstyle N}}
\def\ts#1#2{{\textstyle\frac{#1}{#2}}}
\def\frak g{\mathfrak g}
\def\frak k{\mathfrak k}
\def\raise-1pt\hbox{$\,\stackrel{\wedge}{,}\,$}{\raise-1pt\hbox{$\,\stackrel{\wedge}{,}\,$}}
\def{\rm tr}\,{{\rm tr}\,}
\def\partial {\partial}
\defA{A}
\defB{B}
\defC{C}
\defD{D}
\defE{E}
\defF{F}
\defH{H}
\def\varphi{\varphi}
\setcounter{footnote}0
\begin{document}
\footnotesize
\normalsize
\newpage
\thispagestyle{empty}
\begin{center}
\phantom.
\bigski
{\hfill{\normalsize hep-th/0609082}\\
\hfill{\normalsize HMI-06-03}\\
\hfill{\normalsize ITEP-TH-06-42}\\
\hfill{\normalsize TCD-MATH-06-XX}\\
[10mm]\Large\bf
Baxter $Q$-operator and Givental integral representation for
$C_n$ and $D_n$}
\vspace{0.5cm}
\bigskip\bigskip
{\large A. Gerasimov}
\\ \bigskip
{\it Institute for Theoretical and
Experimental Physics, Moscow, 117259, Moscow, Russia} \\
{\it School of Mathematics, Trinity
College, Dublin 2, Ireland } \\
{\it Hamilton
Mathematics Institute, TCD, Dublin 2, Ireland},\\
\bigskip
{\large D. Lebedev\footnote{E-mail: lebedev@ihes.fr}}
\\ \bigskip
{\it Institute for Theoretical and Experimental Physics, 117259,
Moscow, Russia} \\
{\it l'Institute des Hautes
\'Etudes Scientifiques, 35 route de Chartres,
Bures-sur-Yvette, France},\\
\bigskip
{\large S. Oblezin} \footnote{E-mail: Sergey.Oblezin@itep.ru}\\
\bigskip {\it Institute for Theoretical and Experimental Physics,
117259, Moscow,
Russia}\\
{\it Max-Planck-Institut f\"ur Mathematik, Vivatsgasse 7, D-53111
Bonn, Germany},\\
\end{center}
\vspace{0.5cm}
\begin{abstract}
\noindent
Recently integral representations for the
eigenfunctions of quadratic open Toda chain Hamiltonians
for classical groups was proposed. This representation
generalizes Givental representation for $A_n$.
In this note we verify that the wave functions defined by these
integral representations are common eigenfunctions for the
complete set of open Toda chain Hamiltonians. We consider the zero eigenvalue
wave functions for classical groups $C_n$ and $D_n$ thus completing
the generalization of the Givental construction in these cases.
The construction is based on a recursive procedure
and uses the formalism of Baxter $Q$-operators. We also verify
that the integral $Q$-operators for $C_n$, $D_n$ and twisted affine algebra
$A_{2n-1}^{(2)}$ proposed previously intertwine complete sets of
Hamiltonian operators. Finally we provide integral
representations of the eigenfunctions of the
quadratic $D_n$ Toda chain Hamiltonians for generic nonzero eigenvalues.
\end{abstract}
\vspace{1cm}
\clearpage \newpage
\normalsize
\section{Introduction}
In \cite{GLO} we provided integral representations for
zero eigenvalue wave functions of open Toda chain quadratic Hamiltonian
operators corresponding to the classical series of finite Lie algebras $B_n$, $C_n$,
and $D_n$. This representation generalizes Givental representation for $A_n$ \cite{Gi}.
In this short note we verify the eigenvalue properties for the full set of the
Hamiltonian operators for $C_n$ and $D_n$ open Toda chains.
We also give as an example of the
integral representations for generic eigenvalues
the explicit expression for the wave function of $D_n$ Toda chain.
We use a recursive representation of the
common eigenfunctions of the full sets of $C_n$ and $D_n$ Toda chain
Hamiltonians. The integral recursive operator
is naturally represented as a composition of two elementary integral operators
with simple integral kernels. The interesting property of
these elementary integral operators is that they relate $C_n$ and
$D_n$ wave functions. This is the main reason why we consider these two cases together.
Baxter $Q$-operator is a key tool in the theory of quantum
integrable systems \cite{B}. It plays the role of the generator of quantum
Backlund transformations and was explicitly constructed for $A_{n}^{(1)}$
closed Toda chain in \cite{PG}. It was noted in
\cite{GKLO} that the recursive operators for $A_{n}$ open Toda chain
obtained as a limit of $Q$-operators of $A_{n}^{(1)}$ Toda chain
play the crucial role in the construction of the
Givental integral representation.
In \cite{GLO} we generalize this relation by constructing
$A_{2n-1}^{(2)}$ Baxter $Q$-operator and demonstrating that $C_n$
and $D_n$ recursive operators can be obtained from $A_{2n-1}^{(2)}$
Baxter $Q$-operator in a similar limit. The properties of the
recursive operators are then follows from the commutation
relation between quantum $L$-operator and $Q$-operator
of $A^{(2)}_{2n-1}$ closed Toda chain.
In this note we verify that the integral $Q$-operators for
$A_{2n-1}^{(2)}$ Toda chain introduced in \cite{GLO} commute with
the full set of $A_{2n-1}^{(2)}$ Toda chain Hamiltonians.
The proof is based on the explicit commutation relations with
$L$-operator for Toda chain. In a certain limit this leads
to the commutation relations between recursive operators
and Hamiltonian operators of $C_n$ and $D_n$ open Toda chains.
The fact that the wave functions constructed in \cite{GLO}
are common eigenfunctions of the full set of the Toda chain
then easily follows form the commutation relations.
The use of $A_{2n-1}^{(2)}$ is not very essential in this construction
and is dictated by the simplicity of the resulting degeneration
procedure. For example similar
relations exist between $Q$-operators for $C^{(1)}_n$ and
$D_n^{(1)}$ and recursive operators for $C_n$ and $D_n$. These
constructions will be presented elsewhere together with a complete account of the
constructions of the Baxter $Q$-operators, recursive operators and
integral representations of wave functions for all classical
series \cite{GLO1}.
The plan of this note is as follows. In Section 2
we recall explicit expressions for the Baxter $Q$-operator
for $A_{2n-1}^{(2)}$ given in \cite{GLO} and prove that it commutes with
the full set of $A_{2n-1}^{(2)}$ Toda chain Hamiltonian operators.
In Section 3 we derive recursive operators
for $C_n$ and $D_n$ as a certain limit of the $A_{2n-1}^{(2)}$
Baxter $Q$-operator. We prove the intertwining relations
of the recursive operators with Hamiltonians of
$C_n$ and $D_n$ Toda chains. In Section 4
integral representations of the
eigenfunctions of $D_n$ Toda chain quadratic Hamiltonian
operators for generic eigenvalues are given.
Finally in the Appendix we explicitly check that thus obtained
wave functions for $D_2$ open Toda chain can be reduced
to a product of the wave function of two independent $A_1$
Toda chains. Although it is a consequence of the isomorphism
$D_2=A_1\oplus A_1$ the transformation is rather nontrivial
and serves as an independent check of our construction.
{\em Acknowledgments}: The research of A.~Gerasimov
is partly supported by the grant RFBR 04-01-00646
and Enterprise Ireland Basic Research Grant;
The research of D.~Lebedev is partially supported
by the grant RFBR 04-01-00646.
The research of S. Oblezin was partially supported by the grant RFBR
04-01-00642. D.~Lebedev is grateful to
Institute des Hautes \'Etudes Scientifiques for a warm
hospitality. S.~Oblezin is grateful to Max-Planck-Institut f\"ur Mathematik
for excellent working conditions.
\section{Baxter $Q$-operator for $A_{2n-1}^{(2)}$}
In this section we recall the construction of $Q$-operator
for $A_{2n-1}^{(2)}$ and explicitly check that it commutes with
the full set of $A_{2n-1}^{(2)}$ closed Toda chain Hamiltonians.
Let us start with the description of $A_{2n-1}^{(2)}$ closed
Toda chain. For the relevant facts on the root systems see \cite{K},\cite{DS}. The
detailed description of the corresponding Toda chains can be found
in \cite{RSTS}.
Let us fix an orthonormal basis $\{e_1,\ldots,e_{n}\}$ in
$\mathbb{R}^{n}$. Simple roots of the twisted affine root system $A_{2n-1}^{(2)}$
can be the represented in the following form
\begin{eqnarray}\new\begin{array}{cc} \alpha_1=2e_1,\qquad
\alpha_{i+1}=e_{i+1}-e_i,\qquad 1\leq i\leq n-1\qquad
\alpha_{n+1}=-e_n-e_{n-1}, \end{array}\end{eqnarray}
The corresponding Dynkin diagram is given by
$$
\begin{CD}\alpha_1 \Longrightarrow \alpha_2 @>>> \ldots @>>> \alpha_{n-1}
@>>> \alpha_n\\ @. @. @VVV @.\\ @. @.\alpha_{n+1}@.\end{CD}
$$
To construct $L$-operator of the closed Toda chain
corresponding to an affine root system one should chose an evaluation
representation of the corresponding affine Lie algebra.
Let $\{e_i,h_i,f_i\}$ be a bases of the twisted Lie
algebra $\mathfrak{g}$ corresponding to root system $A_{2n-1}^{(2)}$.
We define the evaluation representation of $\widehat{\mathfrak{g}}$ as follows
(see e.g. \cite{DS}). Consider the following homomorphism
\begin{eqnarray}\new\begin{array}{cc}
\pi:\widehat{\mathfrak{g}}\longrightarrow Mat(2n,\mathbb{C})\otimes \mathbb{C}[u,u^{-1}]
\end{array}\end{eqnarray}
defined explicitly as
\begin{eqnarray}\new\begin{array}{cc}
\pi(e_0)=u(E_{1,2n-1}+E_{2,2n})/2,\qquad \pi(f_0)=2u^{-1}(E_{2n-1,1}+E_{2n,2}),\\
\pi(e_i)=E_{i+1,i}+E_{2n+1-i,2n-i},\qquad \pi(f_i)=E_{i,i+1}+E_{2n-i,2n+1-i},\\
\pi(e_n)=E_{n+1,n},\qquad \pi(f_n)=2E_{n,n+1},\\
\pi(h_0)=(E_{1,1}-E_{2n,2n})+(E_{2,2}-E_{2n-1,2n-1}),\\
\pi(h_i)=(E_{i+1,i+1}-E_{i,i})+(E_{2n+1-i,2n+1-i}-E_{2n-i,2n-i}),\\
\pi(h_n)=2(E_{n+1,n+1}-E_{n,n})\end{array}\end{eqnarray} where
$(E_{ij})_{kl}=\delta_{ik}\delta_{jl}$ in a standard bases in
$Mat(2n,\mathbb{C})$. The evaluation homomorphism then defined by
the evaluation at a particular value of the variable $u$.
Then classical $L$-operator for $A_{2n-1}^{(2)}$ Toda chain has the
following form in this representation \begin{eqnarray}\new\begin{array}{cc}\label{Loperator}
L(p,x;u)\,=\,\sum_{i=1}^n
p_{x_i}\Big(E_{i,i}-E_{2n+1-i,2n+1-i}\Big)\,+
q_i(x)\Big(E_{i,i+1}-E_{2n-i,2n+1-i}\Big)-\\
\Big(E_{i+1,i}-E_{2n+1-i,2n-i}\Big)+
2q_0(x)\Big(E_{2n-1,1}-E_{2n,2}\Big)-\frac{u}{2}
\Big(E_{1,2n-1}-E_{2,2n}\Big),
\end{array}\end{eqnarray}
where $q_i(x)=e^{(\alpha_i,x)}$ and $x=\sum_{i=1}^n x_ie_i$.
The space of $L$-operators (\ref{Loperator})
provides a model for the phase space of the classical $A_{2n-1}^{(2)}$
Toda chain. The quantum ${\cal L}$-operator is obtained by the standard
substitution $p_{x_j}=-i\hbar\partial_{x_j}$.
\begin{eqnarray}\new\begin{array}{cc}\label{qLoperator}
{\cal L}(\partial_x,x;u)\,=\,\sum_{i=1}^n
-i\Big(E_{i,i}-E_{2n+1-i,2n+1-i}\Big)\,\frac{\partial}{\partial x_i}+
q_i(x)\Big(E_{i,i+1}-E_{2n-i,2n+1-i}\Big)-\\
\Big(E_{i+1,i}-E_{2n+1-i,2n-i}\Big)+
2q_0(x)\Big(E_{2n-1,1}-E_{2n,2}\Big)-\frac{u}{2}
\Big(E_{1,2n-1}-E_{2,2n}\Big).
\end{array}\end{eqnarray}
Classical Hamiltonian operators
of the Toda chain are given by the coefficients of the
characteristic polynomial of $L$ \begin{eqnarray}\new\begin{array}{cc} \label{classhamgen}
\det(L(u)-\lambda)=u+\frac{1}{u}+\sum_{i=k}^N \,h_{2k}(p,x)\,
\lambda^{2n-2k} \end{array}\end{eqnarray} For example the quadratic Hamiltonian generator is
given by \begin{eqnarray}\new\begin{array}{cc}
H_2^{A_{2n-1}^{(2)}}(x)=\frac{1}{2}\sum_{i=1}^n\,p_{x_i}^2+e^{2x_1}+
\sum_{i=1}^{n-1}e^{x_{i+1}-x_i}\,+e^{-x_n-x_{n-1}} \end{array}\end{eqnarray}
In \cite{GLO} we introduced the following integral operator
defined by the integral kernel
\bqa\label{inttwA2odd}
Q_{A^{(2)}_{2n-1}}(x_i,\,z_i)&=& \exp (\frac{i}{\hbar}\mathcal{F}(x_i,z_i))=\\
&=&\exp\,\frac{i}{\hbar}\,\Big\{\, g_1e^{x_1+z_1}+ \sum_{i=1}^{n-1}\Big(
e^{x_i-z_i}+g_{i+1}e^{z_{i+1}-x_i}\Big)+
e^{x_n-z_n}+g_{n+1}e^{-x_{n}-z_n}\,\Big\},\nonumber\eqa
The action of the $Q$-operator on the functions is given by
\begin{eqnarray}\new\begin{array}{cc}
(Q_{A_{2n-1}^{(2)}} f)(x_i)=\int \bigwedge_{i=1}^n dz_i
\,\, Q_{A_{2n-1}^{(2)}}(x_i,z_i)\,f(z_i)
\end{array}\end{eqnarray}
Note that $\mathcal{F}$ is a generating function
of the canonical transformation corresponding to the integral
operator in the classical limit.
It was shown in \cite{GLO} that thus defined integral operator
(\ref{inttwA2odd}) intertwines quadratic Hamiltonian operators
$H_2^{A_{2n-1}^{(2)}}$ for different coupling constants
\begin{eqnarray}\new\begin{array}{cc}\label{intertw}
H_2(x_i,\partial_{x_i},g_i)\,Q(x_i,z_i)=Q(x_i,z_i,g_i)H_2(z_i,\partial_{z_i},g_i')
\end{array}\end{eqnarray}
where we use the quadratic Hamiltonian operators
\begin{eqnarray}\new\begin{array}{cc}
H_2^{A_{2n-1}^{(2)}}(x)=\frac{1}{2}\sum_{i=1}^n\,p_{x_i}^2+2g_1e^{2x_1}+
\sum_{i=1}^{n-1}g_i \,e^{x_{i+1}-x_i}\,+g_ng_{n+1}e^{-x_n-x_{n-1}} \end{array}\end{eqnarray}
and the relation holds for $g_i=g'_{n+2-i}=1$, $i\neq 1$ and
$g_1=g'_{n+1}=2$ (see \cite{GLO} for details). In the following
we will assume that these particular coupling constants are chosen.
Also note that in (\ref{intertw}) and in the following similar
identities we shall assume that
the Hamiltonian operator on l.h.s. acts on the right and
the Hamiltonian on r.h.s. acts on the left.
Let us show that the integral operator $Q$ commutes with all Hamiltonian operators
of $A_{2n-1}^{(2)}$ Toda chain. The prove is based on the following
result.
\begin{te}\label{TheoremAn}
Let ${\cal L}(x_i,\partial_{x_i},g_i,u)$ be a quantum $L$-operator given by
(\ref{qLoperator}) and $Q(x_i,z_i)$ be an integral kernel
(\ref{inttwA2odd}). Then the following relation holds
\begin{eqnarray}\new\begin{array}{cc}
R(x_i,z_i,g_i,g'_i,u){\cal L}(x_i,\partial_{x_i},g_i,u)
Q(x_i,z_i,u)=Q(x_i,z_i,u){\cal L}(z_i,\partial_{z_i},g'_i,u) R(x_i,z_i,g_i,g'_i,u)
\nonumber \end{array}\end{eqnarray}
where matrix $R$ is given by
\begin{eqnarray}\new\begin{array}{cc} \label{Rmatrix}
R(x_i,z_i,g_i,g'_i,u)=\Big(e^{x_1+z_1}\,E_{n,1}+\frac{u}{2}E_{1,n}\Big)+
\sum_{i=0}^n\Big(e^{z_i-x_{i-1}}\,E_{n+1,i+1}-uE_{i,n+i}\Big)+\\
\Big(e^{-x_n-z_n}\,E_{2n,n+1}+\frac{u}{2}E_{n+1,2n}\Big)+
\sum_{i=1}^n\Big(e^{x_{n+1-i}-z_{n+1-i}}\,E_{i,n+i}-E_{n+i,i}\Big)\end{array}\end{eqnarray}
\end{te}
The proof is straightforward and reduces to the verification of the
matrix identity \begin{eqnarray}\new\begin{array}{cc} \label{classicalmult}
R(x_i,z_i,g_i,g'_i,u)L(x_i,p_{x_i},g_i,u)
=L(z_i,p_{z_j},g'_i,u)R(x_i,z_i,g_i,g'_i,u)
\end{array}\end{eqnarray} where \bqa\label{cantrans}
p_{x_1}=\frac{\partial \mathcal{F}}{\partial x_1}=
e^{z_1+x_1}+e^{x_1-z_1}-e^{z_2-x_1},\qquad
p_{x_i}=\frac{\partial \mathcal{F}}{\partial x_i}=e^{x_i-z_i}-e^{z_{i+1}-x_i}\,,\\
p_{z_i}=-\frac{\partial \mathcal{F}}{\partial z_i}=
e^{x_i-z_i}-e^{z_i-x_{i-1}},\qquad p_{z_n}=-\frac{\partial
\mathcal{F}}{\partial
z_n}=e^{-x_n-z_n}+e^{x_n-z_n}-e^{z_n-x_{n-1}}\eqa
Note that the identity (\ref{classicalmult}) follows from the following
representations of the pair of $L$-operators
\begin{eqnarray}\new\begin{array}{cc}\label{factone}
L(x_i,p_{x_i},g_i,u)=R\,\,R^*,\,\,\,\,\,
L(z_i,p_{z_i},g'_i,u)=R^*\,\,R, \end{array}\end{eqnarray}
where $R$ is given by (\ref{Rmatrix}) and relations
(\ref{cantrans}) are implied. The involution $R\to R^*$ is the
involution that describes the twisted affine Lie algebra
$A_{2n-1}^{(2)}$ as a fixed point subalgebra of $A_{2n-1}^{(1)}$
(see \cite{DS} for the details). The equations (\ref{factone})
implies that $L$ is in the image of $A_{2n-1}^{(2)}$.
We would like also note that the similar type of relations
was used in the theory of Darboux maps (see \cite{AvM},\cite{V}).
\begin{ex} In the case of $n=4$ adopt the following notations
$$a_i=e^{x_i-z_i},\qquad d_0=e^{x_1+z_1},\quad
d_i=e^{z_i-x_{i-1}},\quad d_n=e^{-x_n-z_n}\,.$$ In these terms we
get for $L(x_i,p_{x_i},g_i,u)$ with $g_i=1$
$$
\left(\begin{array}{cccccccc}
a_4-d_4 & a_4d_3 & 0 & 0 & 0 & 0 & -u/2 & 0\\
-1 & a_3-d_3 & a_3d_2 & 0 & 0 & 0 & 0 & -u/2\\
0 & -1 & a_2-d_2 & a_2d_1 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & a_1+d_0-d_1 & 2a_1d_0 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & -a_1-d_0+d_1 & 2a_2d_1 & 0 & 0\\
0 & 0 & 0 & 0 & -1 & -a_2+d_2 & a_3d_2 & 0\\
2u^{-1}d_3d_4 & 0 & 0 & 0 & 0 & -1 & -a_3+d_3 & a_4d_3\\
0 & 2u^{-1}d_3d_4 & 0 & 0 & 0 & 0 & -1 & -a_4+d_4
\end{array}\right).$$
Then we have
\bqa R=\, \left(\begin{array}{cccccccc}
0 & 0 & 0 & -u/2 & e^{x_4-z_4}u & 0 & 0 & 0\\
0 & 0 & 0 & 0 & -u & e^{x_3-z_3}u & 0 & 0\\
0 & 0 & 0 & 0 & 0 & -u & e^{x_2-z_2}u & 0\\
2e^{z_1+x_1} & 0 & 0 & 0 & 0 & 0 & -u & e^{x_1-z_1}u\\
-1 & e^{z_2-x_1} & 0 & 0 & 0 & 0 & 0 & -u/2\\
0 & -1 & e^{z_3-x_2} & 0 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & e^{z_4-x_3} & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 2e^{-z_4-x_4} & 0 & 0 & 0
\end{array}\right)\eqa and
$$R^*\,=\, \left(\begin{array}{cccccccc}
0 & 0 & 0 & 1/2 & e^{x_1-z_1} & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & e^{x_2-z_2} & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & e^{x_3-z_3} & 0\\
2u^{-1}e^{-z_4-x_4} & 0 & 0 & 0 & 0 & 0 & 1 & e^{x_4-z_4}\\
u^{-1} & u^{-1}e^{z_4-x_3} & 0 & 0 & 0 & 0 & 0 & 1/2\\
0 & u^{-1} & u^{-1}e^{z_3-x_2} & 0 & 0 & 0 & 0 & 0\\
0 & 0 & u^{-1} & u^{-1}e^{z_2-x_1} & 0 & 0 & 0 & 0\\
0 & 0 & 0 & u^{-1} & 2u^{-1}e^{z_1+x_1} & 0 & 0 & 0
\end{array}\right)$$
\end{ex}
Taking into account (\ref{classicalmult}) we have
the following proposition.
\begin{prop} The following identity holds
\bqa\label{detident}
\det(L(x_i,p_{x_i},g_i,u)-\lambda)\,=\,
\det(L(z_i,p_{z_i},g'_i,u)-\lambda)\eqa
\end{prop}
The same reasoning goes for the quantum $L$-operators.
Let $H_i^{A_{2n-1}^{(2)}}(x_i,\partial_{x_i},g_i)$ be a full set of the
commuting quantum Hamiltonian operators for $A_{2n-1}^{(2)}$ Toda chain
given by the quantization of the classical Hamiltonian
operators (\ref{classhamgen}). The quantization uses the
recursive representation for the characteristic polynomial of the
quantum $L$-operator and is based on the special properties of the matrix
differential operator (\ref{qLoperator}).
Then the following intertwining relations holds
\begin{eqnarray}\new\begin{array}{cc}\label{qdetident}
H_i^{A_{2n-1}^{(2)}}(x_i,\partial_{x_i},g_i)Q(x_i,z_i)=
Q(x_i,z_i) H_i^{A_{2n-1}^{(2)}}(z_i,\partial_{z_i},g'_i) \nonumber
\end{array}\end{eqnarray}
Let us stress that the constructed integral operators
intertwine the Hamiltonians of the $A_{2n-1}^{(2)}$ Toda chain
with different coupling constants. It was explained in \cite{GLO} that
to get the Baxter $Q$-operator for $A_{2n-1}^{(2)}$ commuting with the
Hamiltonians one should apply the elementary intertwining twice
\bqa
Q^{A^{(2)}_{2n-1}}(x_1,\ldots,x_n;\,y_1,\ldots,y_n)=\int
\bigwedge_{i=1}^{n}dz_i\,
Q(x_1,\ldots,x_n;\,z_1,\ldots,z_{n})\times
\\ \nonumber
\times Q(y_1,\ldots,y_{n};\,z_1,\ldots,z_n).\eqa
\section{Recursive operators for $C_n$ and $D_n$}
In this section we prove the intertwining
relations between recursive operators for
the full set of Hamiltonian operators for
for $C_n$ and $D_n$ open Toda chains.
These commutation relations can be also obtained
from the analogous properties of Baxter $Q$-operator
of $A_{2n-1}^{(2)}$ Toda chain in a certain limit.
We start with the description of $C_n$ and $D_n$ Toda chains.
The root system $C_n$ and $D_n$ and the Hamiltonian
are defined in terms of the orthonormal basis $\{e_1,\ldots,e_{n}\}$ in
$\mathbb{R}^{n}$ as follows.
Simple roots of the root system $C_n$ are given by \begin{eqnarray}\new\begin{array}{cc} \alpha_1=2e_1,\qquad
\alpha_{i+1}=e_{i+1}-e_i,\qquad 1\leq i\leq n-1\qquad
\alpha_{n+1}=-e_n-e_{n-1}, \end{array}\end{eqnarray}
The corresponding Dynkin diagram is
$$
\begin{CD}\alpha_1 \Longrightarrow \alpha_2 @>>> \ldots @>>> \alpha_{n-1}
@>>> \alpha_n\\ @. @. @VVV @.\\ @. @.\alpha_{n+1}@.\end{CD}
$$
Simple roots of the root system $D_n$ are given by \begin{eqnarray}\new\begin{array}{cc} \alpha_1=2e_1,\qquad
\alpha_{i+1}=e_{i+1}-e_i,\qquad 1\leq i\leq n-1\qquad
\alpha_{n+1}=-e_n-e_{n-1}, \end{array}\end{eqnarray}
The corresponding Dynkin diagram is
$$
\begin{CD}\alpha_1 \Longrightarrow \alpha_2 @>>> \ldots @>>> \alpha_{n-1}
@>>> \alpha_n\\ @. @. @VVV @.\\ @. @.\alpha_{n+1}@.\end{CD}
$$
Classical Hamiltonian operators $h^{C_n}_i$ of $C_n$ Toda chain are given by the
coefficients of the characteristic polynomial
\begin{eqnarray}\new\begin{array}{cc} \label{charpol}
\det(L(x,p)-\lambda)=\sum_{i} h_i\lambda^{n-i}
\end{array}\end{eqnarray}
where $L$ operator is given by
\begin{eqnarray}\new\begin{array}{cc}
L^{C_n}(x,p)=\sum_{i=1}^n\,p_i\Big(E_{i,i}-E_{2n+1-i,2n+1-i}\Big)+\\
q_i(x)\Big(E_{i,i+1}+E_{2n-i,2n+1-i}\Big)-
\Big(E_{i+1,i}-E_{2n+1-i,2n-i}\Big) \end{array}\end{eqnarray}
Similarly classical Hamiltonian operators of $D_n$ Toda chain are
given by the coefficients of characteristic polynomial (\ref{charpol})
of the following $L$-operator
\begin{eqnarray}\new\begin{array}{cc}
L^{D_n}(x,p)=
\sum_{i=1}^n\,p_i\Big(E_{i,i}-E_{2n+1-i,2n+1-i}\Big)+\\
\sum_{i=1}^{n-1}\,q_i(x)\Big(E_{i,i+1}+E_{2n-i,2n+1-i}\Big)-
\Big(E_{i+1,i}-E_{2n+1-i,2n-i}\Big)+\\
2q_n(x)\Big(E_{n-1,n+1}+E_{n,n+2}\Big)-(E_{n+1,n-1}+E_{n+2,n}) \end{array}\end{eqnarray}
The corresponding quantum Hamiltonians are obtained by the standard
quantisation procedure. Thus for the quadratic Hamiltonian operators
we have
\begin{eqnarray}\new\begin{array}{cc} H_2^{C_n}(z_i)=-\frac{1}{2}\sum_{i=1}^n
\frac{\partial^2}{\partial z_i^2}\,+\sum_{i=1}^{n-1}
e^{z_{i+1}-z_i}\,+e^{-2z_n},\\
H_2^{D_n}(x_i)= -\frac{1}{2}\sum_{i=1}^n \frac{\partial^2}{\partial
x_i^2}\,+\sum_{i=1}^{n-1} e^{x_{i+1}-x_i}\,+e^{-z_n-z_{n-1}}\end{array}\end{eqnarray}
In \cite{GLO} the integral operators intertwining
quadratic Toda chain Hamiltonian operators for different
elements of the classical series $C_n$ and $D_n$ were proposed.
Integral operator defined by the kernel \begin{eqnarray}\new\begin{array}{cc}\label{DnQCn}
Q^{D_{n}}_{C_{n}}(x_i;z_i)\,=\exp\, \frac{i}{\hbar}\mathcal{F}^{D_{n}}_{C_{n}}(x_i;z_i)
=\exp\,\frac{i}{\hbar}\Big\{\,\sum_{i=1}^{n-1}\Big(e^{x_i-z_i}+
e^{z_{i+1}-x_i}\Big)\,+\,e^{x_n-z_n}+e^{-x_n-z_n}\,\Big\},\end{array}\end{eqnarray}
satisfies the following intertwining relation
\begin{eqnarray}\new\begin{array}{cc} \label{HamintCn}
H_2^{C_n}(x_i,\partial_{x_i})\,Q_{D_{n}}^{C_{n}}(x_i,z_i)=
Q_{D_{n}}^{C_{n}}(x_i,z_i)H_2^{D_n}(z_i,\partial_{z_i}).
\end{array}\end{eqnarray}
Similarly integral operator defined by the following kernel \begin{eqnarray}\new\begin{array}{cc} \label{DnCnminus}
Q_{D_n}^{\,\,\,C_{n-1}}(x_i;z_i)=\exp\,\frac{i}{\hbar}
\mathcal{F}_{D_{n}}^{C_{n-1}}(x_i;z_i)=
\exp\,\frac{i}{\hbar}\Big\{\sum_{i=1}^{n-1}\Big(e^{z_i-x_i}+g_ie^{x_{i+1}-z_i}\Big)+
g_ne^{-x_n-z_{n-1}}\Big\},\end{array}\end{eqnarray}
satisfies the following intertwining relation
\begin{eqnarray}\new\begin{array}{cc} \label{HamintDn}
H_2^{D_n}(x_i,\partial_{x_i})\,Q_{D_{n}}^{C_{n-1}}(x_i,z_i)=
Q_{D_{n}}^{C_{n-1}}(x_i,z_i)H_2^{C_{n-1}}(z_i,\partial_{z_i})
\end{array}\end{eqnarray}
In the classical approximation (\ref{HamintCn})
is reduced to the identity
\begin{eqnarray}\new\begin{array}{cc}
H_2^{C_n}(x_i,p_{x_i})\,=H_2^{D_n}(z_i,p_{z_i})
\end{array}\end{eqnarray}
where the following expressions for the momenta variables
are implied
\begin{eqnarray}\new\begin{array}{cc}\label{CanTransf}
p_{x_i}=\frac{\partial \mathcal{F}^{D_{n}}_{C_{n}}} {\partial x_i}=
e^{x_i-z_i}-e^{z_{i+1}-x_i},\\ p_{x_n}=\frac{\partial
\mathcal{F}^{D_{n}}_{C_{n}}} {\partial
x_n}=e^{x_n-z_n}-e^{-z_{n}-x_n}\qquad 1\leq i< n,\end{array}\end{eqnarray} \begin{eqnarray}\new\begin{array}{cc}
p_{z_1}=-\frac{\partial \mathcal{F}^{D_{n}}_{C_{n}}}{\partial z_1}=
e^{x_1-z_1},\qquad p_{z_i}=-\frac{\partial
\mathcal{F}^{D_{n}}_{C_{n}}}{\partial z_i}=
e^{x_i-z_i}-e^{z_i-x_{i-1}},\\ p_{z_n}=-\frac{\partial
\mathcal{F}^{D_{n}}_{C_{n}}}{\partial z_n}=
e^{x_n-z_n}+e^{-z_n-x_n}-e^{z_n-x_{n-1}},\qquad 1<i<n.\end{array}\end{eqnarray}
Similarly (\ref{HamintDn}) is reduced in the classical approximation
to \begin{eqnarray}\new\begin{array}{cc}
H_2^{D_n}(x_i,p_{x_i})\,= H_2^{C_{n-1}}(z_i,p_{z_i}) \end{array}\end{eqnarray} where the
following expressions for the momenta variables are implied \begin{eqnarray}\new\begin{array}{cc}
\label{CanTransfone}
p_{x_1}=\frac{\partial \mathcal{F}_{D_{n}}^{C_{n-1}}}
{\partial x_1}=-e^{z_1-x_1},\qquad p_{x_i}=\frac{\partial
\mathcal{F}_{D_{n}}^{C_{n-1}}}{\partial x_i}=
e^{x_i-z_{i-1}}-e^{z_i-x_i},\\ p_{x_n}=\frac{\partial
\mathcal{F}_{D_{n}}^{C_{n-1}}}{\partial x_n}=
e^{x_n-z_{n-1}}-e^{-z_{n-1}-x_n},\qquad 1<i<n,\end{array}\end{eqnarray} \begin{eqnarray}\new\begin{array}{cc}
p_{z_i}=-\frac{\partial\mathcal{F}_{D_{n}}^{C_{n-1}}}{\partial
z_i}=e^{x_{i+1}-z_i}-e^{z_i-x_i},\\ p_{z_{n-1}}=\frac{\partial
\mathcal{F}_{D_{n}}^{C_{n-1}}}{\partial
z_{n-1}}=e^{-z_{n-1}-x_n}+e^{x_n-z_{n-1}}-e^{z_{n-1}-x_{n-1}},\qquad
1\leq i<n-1.\end{array}\end{eqnarray}
Toda chains corresponding to the root systems $C_n$ and $D_n$ can be
obtained from $A_{2n-1}^{(2)}$ Toda chain using an appropriate
limiting procedure. To do this explicitly one should
explicitly introduce the coupling constant in Toda chains.
It was shown in \cite{GLO} that the operator \bqa
Q_{A^{(2)}_{2n-1}}(x_i,\,z_i)&=&\frac{i}{\hbar} \exp \mathcal{F}(x_i,z_i)=\\
&=&\exp \,\frac{i}{\hbar}\Big\{\, g_1e^{x_1+z_1}+ \sum_{i=1}^{n-1}\Big(
e^{x_i-z_i}+g_{i+1}e^{z_{i+1}-x_i}\Big)+
e^{x_n-z_n}+g_{n+1}e^{-x_{n}-z_n}\,\Big\}\nonumber\eqa satisfies the
following relation
\begin{eqnarray}\new\begin{array}{cc} H_2^{A_{2n-1}^{(2)}}(x_i)Q_{A^{(2)}_{2n-1}}(x_i,\,z_i)\,=\,
Q_{A^{(2)}_{2n-1}}(x_i,\,z_i)\widetilde{H}_2^{A_{2n-1}^{(2)}}(z_i)
\end{array}\end{eqnarray}
for the quadratic Hamiltonians associated to the root system of type
$A_{2n-1}^{(2)}$:
\begin{eqnarray}\new\begin{array}{cc} H_2^{A_{2n-1}^{(2)}}(x_i)= -\frac{1}{2}\sum_{i=1}^n
\frac{\partial^2}{\partial x_i^2}\,+2g_1e^{2x_1}+\sum_{i=1}^{n-1}
g_{i+1}e^{x_{i+1}-x_i}\,+g_ng_{n+1}e^{-x_n-x_{n-1}}
\end{array}\end{eqnarray} and
\begin{eqnarray}\new\begin{array}{cc}
\widetilde{H}_2^{A_{2n-1}^{(2)}}(z_i)= -\frac{1}{2}\sum_{i=1}^n
\frac{\partial^2}{\partial z_i^2}\,+ g_1g_2e^{z_1+z_2}+
\sum_{i=1}^{n-1} g_{i+1}e^{z_{i+1}-z_i}\,+2g_{n+1}e^{-2z_n}
\end{array}\end{eqnarray}
Taking the limit $g_1\rightarrow0$ we obtain that $Q$-operator
\begin{eqnarray}\new\begin{array}{cc}
Q^{D_{n}}_{C_{n}}(x_i;z_i)\,=\,\lim_{g_1\rightarrow0}
Q_{A^{(2)}_{2n-1}}(x_i,z_i)
\end{array}\end{eqnarray} intertwines
\begin{eqnarray}\new\begin{array}{cc} H_2^{C_n}(z_i)\,=\,\lim_{g_1\rightarrow0}
\widetilde{H}_2^{A_{2n-1}^{(2)}}(z_i)
\end{array}\end{eqnarray} and
\begin{eqnarray}\new\begin{array}{cc} H_2^{D_n}(x_i)\,=\,\lim_{g_1\rightarrow0}
H_2^{A_{2n-1}^{(2)}}(x_i)\,.
\end{array}\end{eqnarray} These Hamiltonians are naturally
associated with root systems of type $C_n$ and $D_n$ respectively that
are isomorphic to certain root sub-systems in the affine root system
of type $A_{2n-1}^{(2)}$. Thus this limit procedure
provides the $L$-operators associated to the root sub-systems of
types $C_n$ and $D_n$.
Note that the limiting operator $Q^{D_{n}}_{C_{n}}(x_i;z_i)$
in general intertwines Hamiltonians of Toda chains
with different coupling constants except last ones. Thus the following
operators :
\begin{eqnarray}\new\begin{array}{cc} H_2^{C_n}(z_i)=-\frac{1}{2}\sum_{i=1}^n
\frac{\partial^2}{\partial z_i^2}\,+\sum_{i=1}^{n-1}
g'_{i+1}e^{z_{i+1}-z_i}\,+2g_ne^{-2z_n}
\end{array}\end{eqnarray} and
\begin{eqnarray}\new\begin{array}{cc} H_2^{D_n}(x_i)=-\frac{1}{2}\sum_{i=1}^n
\frac{\partial^2}{\partial x_i^2}\,+\sum_{i=1}^{n-1}
g_{i+1}e^{x_{i+1}-x_i}\,+g_{n-1}g_ne^{-z_n-z_{n-1}}
\end{array}\end{eqnarray}
are intertwined by the integral operator with the kernel
\bqa Q^{D_{n}}_{C_{n}}(x_i;z_i)\,=\,
\exp\Big\{\,\sum_{i=1}^{n-1}\Big(\gamma_ie^{x_i-z_i}+
\beta_{i}e^{z_{i+1}-x_i}\Big)\,+\,
e^{x_n-z_n}+g_ne^{-x_n-z_n}\,\Big\}\nonumber\eqa where
\begin{eqnarray}\new\begin{array}{cc} \gamma_i=\prod_{k=i}^{n-1}\frac{g'_k}{g_k},\qquad
\beta_i=g_i\prod_{k=i+1}^{n-1}\frac{g_k}{g'_k}\qquad 1\leq i<n\,.
\end{array}\end{eqnarray}
Taking the limit $g'_1\rightarrow0$ we obtain
\begin{eqnarray}\new\begin{array}{cc} H_2^{C_{n-1}}(z_i)\,=\,\lim_{g'_1\rightarrow0}H_2^{C_n}(z_i)
\end{array}\end{eqnarray}
Thus the intertwiner
\begin{eqnarray}\new\begin{array}{cc} Q^{D_n}_{\,\,\,C_{n-1}}(x_i;z_i)\,=\,
\lim_{g'_1\rightarrow0}Q^{D_{n}}_{C_{n}}(x_i;z_i)
\end{array}\end{eqnarray} satisfying
\begin{eqnarray}\new\begin{array}{cc} H_2^{D_n}(x_i)\,Q^{D_{n}}_{C_{n-1}}(x_i,z_i)=
Q^{D_{n}}_{C_{n-1}}(x_i,z_i)H_2^{C_{n-1}}(z_i)
\end{array}\end{eqnarray}
Therefore all the recursive operators can be obtained
from $A_{2n-1}^{(2)}$ Baxter $Q$-operators in different limits.
This implies that the generalization of the intertwining relations
(\ref{HamintCn}) and (\ref{HamintDn}) to the
full set of the Hamiltonians of $C_n$ and $D_n$ open Toda chains
can be obtained from (\ref{detident}), (\ref{qdetident})
using the limiting procedure described above.
The following commutation relations with $L$ operators
are the analog of the Theorem \ref{TheoremAn}.
\begin{te} Let ${\cal L}^{C_n}(x_i,\partial_{x_i},g_i,u)$ be a quantum $L$-operator
of $C_n$ Toda chain and
\\ let ${\cal L}^{D_n}(x_i,\partial_{x_i},g_i,u)$ be a quantum $L$-operator
of $D_n$ Toda chain. Then the following intertwining relations for the
kernels (\ref{DnQCn}),(\ref{DnCnminus}) of the integral operators hold
\bqa \label{basicidentCD}
M(x_i,z_i,g_i,g'_i,u){\cal L}^{C_n}(x_i,\partial_{x_i},g_i,u)
Q^{D_{n}}_{C_{n}}
(x_i,z_i,u)=\\
=Q^{D_{n}}_{C_{n}}
(x_i,z_i){\cal L}^{D_n}(z_i,\partial_{z_i},g'_i,u) M(x_i,z_i,g_i,g'_i,u)
\nonumber \eqa
\bqa \label{basicidentDC}
N(x_i,z_i,g_i,g'_i,u){\cal L}^{D_n}(x_i,\partial_{x_i},g_i,u)
Q_{D_n}^{C_{n-1}}(x_i,z_i,u)=\\
= Q_{D_n}^{C_{n-1}}(x_i,z_i){\cal L}^{C_{n-1}}(z_i,\partial_{z_i},g'_i,u) N(x_i,z_i,g_i,g'_i,u)
\nonumber \eqa
where matrices $M$ and $N$ are given
\begin{eqnarray}\new\begin{array}{cc}
M(x_i,z_i,g_i,g'_i,u)=\sum_{i=1}^n\Big(E_{i,i}+e^{z_i-x_{i-1}}\,E_{i,i+1}\Big)+\\
\Big(e^{x_{n+1-i}-z_{n+1-i}}\,E_{2n+1-i,2n+1-i}+E_{2n+1-i,2n-i}\Big)
+\Big(e^{-x_n-z_n}E_{n,n+1}-\frac{1}{2}E_{n+1,n}\Big)
\end{array}\end{eqnarray} \begin{eqnarray}\new\begin{array}{cc}
N(x_i,z_i,g_i,g'_i,u)=\sum_{i=1}^n\Big(e^{x_i-z_i}\,E_{i,i}-E_{i+1,i}\Big)\\
\Big(E_{2n+1-i,2n+1-i}-e^{z_{n+1-i}-z_{n-i}}\,E_{2n-i,2n+1-i}\Big)
+\Big(e^{-x_n-z_n}E_{n,n+1}+\frac{1}{2}E_{n+1,n}\Big)
\end{array}\end{eqnarray}
\end{te}
Similar to the case of $A_{2n-1}^{(2)}$ the commutation
relations (\ref{basicidentCD}),(\ref{basicidentDC})
provide the quantum intertwining relation
with $C_n$ and $D_n$ Hamiltonian operators.
This leads to the proof of the zero eigenvalue property for the full sets of the
Hamiltonian operators applied to the corresponding wave functions \cite{GLO}.
\section{Integral representation of generic $D_n$ eigenfunctions}
In \cite{GLO} integral representations for
zero eigenvalue wave functions of $D_n$ Toda chain quadratic Hamiltonian
operators were given. In this Section we generalize these representations
to the case of generic eigenvalues.
Following \cite{GLO} we represent the wave functions of $D_n$ Toda
chain using the recursive integral operators. Let us define an
integral recursive operator by the following kernel
\bqa\label{intrepnonzero}
Q_{D_{k+1}}^{\,\,\,\,D_k}(x_{k+1,i},\,x_{k,j},\,\lambda_{k+1})=\int\bigwedge_{i=1}^k\,dz_{k,i}
\exp\Big\{\imath\lambda_{k+1}(\sum_{i=1}^{k+1} x_{k+1,i}+\sum_{i=1}^{k}x_{k,i}-
2\sum_{i=1}^kz_{k,i})\Big\}\,\times \nonumber \\ \times
\left(e^{-x_{k+1,k+1}}+e^{-x_{k,k}}\right)^{2\imath\,\lambda_{k+1}}
\times\\ \times Q_{D_{k+1}}^{\,\,\,\,C_k}(x_{k+1,1},\ldots,x_{k+1,k+1};
z_{k,1},\ldots,z_{k,k})\, Q_{C_{k}}^{\,\,\,\,D_k}
(z_{k,1},\ldots,z_{k,k};x_{k,1},\ldots,x_{k,k}),\nonumber \eqa
where
\begin{eqnarray}\new\begin{array}{cc} Q_{D_{k+1}}^{\,\,\,C_{k}}(x_{k+1,i},\, z_{k,i})=
\exp\Big\{\sum_{i=1}^{k}\Big(e^{z_{k,i}-x_{k+1,i}}+e^{x_{k+1,i+1}-z_{k,i}}\Big)+
e^{-x_{k+1,k+1}-z_{k,k}}\Big\},\end{array}\end{eqnarray}
\begin{eqnarray}\new\begin{array}{cc}
Q_{C_k}^{\,\,\,\,D_{k}}(z_{k,i},\,x_{k,i})=\exp\Big\{\,
\sum_{i=1}^{k}\Big(e^{x_{k,i}-z_{k,i}}+e^{z_{k,i+1}-x_{k,i}}\Big)+
e^{x_{k,k}-z_{k,k}}+e^{-x_{k,k}-z_{k,k}}\Big\}.\end{array}\end{eqnarray}
One can straightforwardly
verify that the following intertwining
relations holds
\begin{eqnarray}\new\begin{array}{cc}
H^{D_{k+1}}(x_{k+1,j})\,\,Q_{D_{k+1}}^{\,\,\,\,D_k}(x_{k+1,j},\,x_{k,j},\,\lambda_{k+1})=
Q_{D_{k+1}}^{\,\,\,\,D_k}(x_{k+1,j},\,x_{k,j},\,\lambda_{k+1})\,
(H^{D_k}(x_{k,j})+\frac{1}{2}\lambda_{k+1}^2), \nonumber
\end{array}\end{eqnarray}
where quadratic Hamiltonian operators are given by
\begin{eqnarray}\new\begin{array}{cc}
H^{D_k}(x_{k,j})=-
\frac{1}{2}\sum_{j=1}^k\frac{\partial^2}{\partial x_{k,j}^2}+
\sum_{i=1}^{k-1} e^{x_{k,j+1}-x_{k,j}}+e^{-x_{k,k}-x_{k,k-1}}.
\end{array}\end{eqnarray}
Consider the wave function for $D_n$ Toda chain satisfying
the eigenfunction equation
\begin{eqnarray}\new\begin{array}{cc}
H^{D_n}(x)\,\,\,\Psi^{D_n}_{\lambda_1,\cdots ,\lambda_n} (x_1,\ldots,x_n)=
\frac{1}{2}(\sum_{i=1}^n\lambda_i^2)\,\,\,
\Psi^{D_n}_{\lambda_1,\cdots ,\lambda_n} (x_1,\ldots,x_n).
\end{array}\end{eqnarray}
Then it can be represented in the following integral from
\begin{eqnarray}\new\begin{array}{cc}\label{WDN}
\Psi^{D_n}_{\lambda_1,\cdots ,\lambda_n} (x_1,\ldots,x_n)\,=\,
\int\bigwedge_{k=1}^{n-1}\bigwedge_{i=1}^kdx_{k,i}\,\,
e^{\imath\lambda_1 x_{11}}\,\,
\prod_{k=1}^{n-1}Q_{D_{k+1}}^{\,\,\,\,D_k}(x_{k+1,j},\,x_{k,j},\,\lambda_{k+1}),\end{array}\end{eqnarray}
where $x_i:=x_{n,i}$.
Let us note that there is another form of the integral representation
for the kernel of the recursive operator (\ref{intrepnonzero})
\begin{eqnarray}\new\begin{array}{cc}
Q_{D_{k+1}}^{\,\,\,\,D_k}(x_{k+1,j},\,x_{k,j},\,\lambda_{k+1})=
\frac{1}{\Gamma(2\imath \,\lambda_{k+1})}\,
\int\bigwedge_{i=1}^k\,dz_{k,i}
\,\, d y_k\,\,\times \\ \times
\exp\Big\{\imath\lambda_{k+1}(\sum_{i=1}^{k+1} x_{k+1,i}+\sum_{i=1}^{k}x_{k,i}-
2\sum_{i=1}^kz_{k,i}+2y_{k+1})\Big\}\,\times \\
\times Q_{D_{k+1}}^{\,\,\,\,C_k}(x_{k+1,1},\ldots,x_{k+1,k+1};
z_{k,1},\ldots,z_{k,k};y_k)\, Q_{C_{k}}^{\,\,\,\,D_k}
(z_{k,1},\ldots,z_{k,k};x_{k,1},\ldots,x_{k,k};y_k).
\end{array}\end{eqnarray}
where
\begin{eqnarray}\new\begin{array}{cc} Q_{D_{k+1}}^{\,\,\,C_{k}}(x_{k+1,i};\, z_{k,i},y_k)=
\exp\Big\{\sum_{i=1}^{k}\Big(e^{z_{k,i}-x_{k+1,i}}+e^{x_{k+1,i+1}-z_{k,i}}\Big)+
e^{-x_{k+1,k+1}-z_{k,k}}+e^{-x_{k+1,k+1}-y_k}\Big\},\nonumber \end{array}\end{eqnarray}
\begin{eqnarray}\new\begin{array}{cc}
Q_{C_k}^{\,\,\,\,D_{k}}(z_{k,i};x_{k,i},y_k)=\exp\Big\{\,
\sum_{i=1}^{k}\Big(e^{x_{k,i}-z_{k,i}}+e^{z_{k,i+1}-x_{k,i}}\Big)+
e^{x_{k,k}-z_{k,k}}+e^{-x_{k,k}-z_{k,k}}+e^{-x_{k,k}-y_k}\Big\}.\nonumber
\end{array}\end{eqnarray}
This integral representation can be easily described in terms of the
appropriate Givental diagram. The detailed discussion of the relation
with the geometry of $D_n$ flag spaces and the results for other
classical finite and affine Lie algebras will be
presented in \cite{GLO1}.
\section{Appendix: Wave function for $D_2$}
Open Toda chain for the root system $D_2$ can be represented as a
couple of non-interacting $A_1$ Toda chains. This is a consequence of the
isomorphism $D_2=A_1\oplus A_1$. Quadratic Hamiltonian
operator of $D_2$ Toda chain is given by
\begin{eqnarray}\new\begin{array}{cc} \label{d2quadratic}
H^{D_2}(x)=-\frac{1}{2}\sum\limits_{i=1}^{2}
\frac{\partial^2}{\partial x_i^2}+
e^{x_{22}-x_{21}}+e^{-x_{22}-x_{21}} \end{array}\end{eqnarray}
Changing the variables \begin{eqnarray}\new\begin{array}{cc} \xi=x_{22}-x_{21},\,\,\,\,\,\, \eta =
x_{22}+x_{21}, \end{array}\end{eqnarray} one transforms $H^{D_2}$ into the
sum of two $A_1$ Hamiltonians: \begin{eqnarray}\new\begin{array}{cc} \label{dtwo} H^{D_2}(x)=
2\,H^{A_1}(\xi)+2\,H^{A_1}(\eta):=
2 \Big(-\frac{1}{2} \frac{\partial^2}{\partial \xi^2}+
\frac{1}{2}e^{\xi}\Big)+2\Big(-\frac{1}{2} \frac{\partial^2}{\partial \eta^2}+
\frac{1}{2}e^{-\eta}\Big).\end{array}\end{eqnarray}
The additional quartic $D_2$ Hamiltonian commuting with
(\ref{d2quadratic}) is given by $H_4^{D_2}(x_1,x_2)=H^{A_1}(\xi)\,H^{A_1}(\eta)$.
Let $\Psi_{\lambda_1,\lambda_2}(x_{21},x_{22})$ be an eigenfunction
of the quadratic $D_2$ Hamiltonian
\begin{eqnarray}\new\begin{array}{cc}
H^{D_2}(x)\,\,\Psi^{D_2}_{\lambda_1,\lambda_2}(x)=(\frac{1}{2}\lambda_1^2+\frac{1}{2}\lambda_2^2)
\,\,\Psi^{D_2}_{\lambda_1,\lambda_2}(x)
\end{array}\end{eqnarray}
Taking into account (\ref{dtwo}) one can represent the eigenfunction
as a product of two eigenfunctions of $A_1$ Hamiltonians as
\begin{eqnarray}\new\begin{array}{cc} \label{factorisation}
\Psi_{\lambda_1,\lambda_2}(x_{21},x_{22})=\chi_{\frac{\lambda_2-\lambda_1}{2}}(-x_{22}-x_{21})\,
\chi_{\frac{\lambda_2+\lambda_1}{2}}(x_{22}-x_{21}),
\end{array}\end{eqnarray}
where $\chi_{\nu}(y)$ satisfies the equation
\begin{eqnarray}\new\begin{array}{cc} \label{aone}
\Big(-\frac{1}{2} \frac{\partial^2}{\partial y^2}+
\frac{1}{2}e^{y}\Big)\chi_{\nu}(y)=\frac{1}{2}\nu^2\chi_{\nu}(y).
\end{array}\end{eqnarray}
Solutions of (\ref{aone}) can expressed through Macdonald function as follows
\begin{eqnarray}\new\begin{array}{cc}
\chi_{\nu}(y)=K_{2\imath\nu}(2e^y)=-\frac{e^{2\pi \nu}
}{2}\int\limits_{-\infty+\imath\pi}^{\infty+\imath\pi}
e^{2\imath\nu(x+y)}\exp\Big\{e^{-x}+e^{x+2y}\Big\}d x.
\end{array}\end{eqnarray}
Below we explicitly demonstrate the factorisation
(\ref{factorisation}) using the integral representation
for $D_2$ Toda chain wave function described in the previous Section.
The eigenfunction of $H^{D_2}$ has the following integral
representation (see (\ref{WDN}) for $n=2$))
\begin{eqnarray}\new\begin{array}{cc}\label{psi} \Psi^{D_2}_{\lambda_1,\lambda_2}(x_{21},x_{22})=\int\int d
x_{11}d z_{11} dy \,\,\exp\Big\{
\imath\lambda_2(x_{22}+x_{21})-\imath\,2\,\lambda_2(z_{11}-y)+\imath(\lambda_2+\lambda_1)x_{11}\Big\}\times
\\ \times
\exp\Big\{e^{z_{11}-x_{21}}+e^{x_{22}-z_{11}}+e^{-x_{22}-z_{11}}
+e^{x_{11}-z_{11}}+e^{-x_{11}-z_{11}}+
e^{-x_{22}-y}+e^{-x_{11}-y}\Big\}.
\end{array}\end{eqnarray}
One can integrate explicitly over $y$ to get the following
representation
\begin{eqnarray}\new\begin{array}{cc}
\label{psinoy} \Psi^{D_2}_{\lambda_1,\lambda_2}(x_{21},x_{22})=\Gamma(2\imath \lambda_2)\,\int\int d
x_{11}d z_{11} \,\,\left(e^{-x_{22}}+e^{-x_{11}}\right)^{\imath
\,2\,\lambda_2}\times \\ \times \exp\Big\{
\imath\lambda_2(x_{22}+x_{21})-\imath \,2\,\lambda_2\,z_{11}+\imath(\lambda_2+\lambda_1)x_{11}\Big\}\times
\\ \times
\exp\Big\{e^{z_{11}-x_{21}}+e^{x_{22}-z_{11}}+e^{-x_{22}-z_{11}}
+e^{x_{11}-z_{11}}+e^{-x_{11}-z_{11}}\Big\}.
\end{array}\end{eqnarray}
Let us introduce new variables
\begin{eqnarray}\new\begin{array}{cc}
\xi=x_{22}-x_{21},\,\,\,\,\,\, \eta =
x_{22}+x_{21},\,\,\,\, z'_{11}=z_{11}-\frac{x_{22}+x_{21}}{2},
\,\,\,\,x'_{11}=x_{11}-\frac{x_{22}+x_{21}}{2} .\end{array}\end{eqnarray}
Then \begin{eqnarray}\new\begin{array}{cc} \Psi^{D_2}_{\lambda_1,\lambda_2}(x_{21},x_{22})=\Gamma(2\imath \lambda_2)\,\int\int d x_{11}' d
z_{11}'\,\,\left(e^{-x_{22}}+e^{-x'_{11}+\frac{\eta}{2}}\right)^{\imath
\,2\,\lambda_2}\times \\ \times \exp\Big\{
-\imath\frac{(\lambda_2+\lambda_1)}{2}\eta-\imath \,2\,\lambda_2\,z'_{11}+\imath(\lambda_2+\lambda_1)x'_{11}\Big\}\times
\\ \times
\exp\Big\{
e^{z_{11}'+\frac{\xi}{2}}+e^{\frac{\xi}{2}-z_{11}'}+e^{x_{11}'-z_{11}'}+
e^{-\frac{\xi}{2}-\eta-z_{11}'}+e^{-x_{11}'-z_{11}'-\eta}\Big\}.
\end{array}\end{eqnarray}
Define new variable as follows \begin{eqnarray}\new\begin{array}{cc}\label{anz}
e^{\tau}=e^{-z_{11}'-\eta}\Big(e^{-\frac{\xi}{2}}+e^{-x_{11}'}\Big),
\end{array}\end{eqnarray}
\begin{eqnarray}\new\begin{array}{cc}
e^{t}=e^{x'_{11}-z_{11}'-\frac{\xi}{2}-\eta}
\Big(e^{-\frac{\xi}{2}}+e^{-x_{11}'}\Big) .
\end{array}\end{eqnarray}
As a result we obtain the following integral representation
\begin{eqnarray}\new\begin{array}{cc} \Psi^{D_2}_{\lambda_1,\lambda_2}(x_{21},x_{22})=\Gamma(2\imath \lambda_2)\,
\int\int d\tau d t \,\,
\exp\Big\{
-\imath\frac{(\lambda_2-\lambda_1)}{2}\eta+\imath \,(\lambda_2-\lambda_1)+\\+
\imath\frac{(\lambda_2+\lambda_1)}{2}\xi + \imath(\lambda_2+\lambda_1)t\Big\}
\exp \Big\{e^{-\tau}+e^{\tau-\eta}+e^{-t}
+e^{t+\xi}\Big\}=\\=4e^{-2\pi \lambda_2}\,\Gamma(2\imath \lambda_2)\,\,
K_{\imath\frac{\lambda_2+\lambda_1}{2}}(2e^{\frac{\xi}{2}})
K_{\imath\frac{\lambda_2-\lambda_1}{2}}(2 e^{-\frac{\eta}{2}}).
\end{array}\end{eqnarray}
Thus we have demonstrated that the proposed integral representation
for $D_n$ Toda chain wave functions is compatible with the factorization
of $D_2$ Toda chain.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,963 |
FSANZ Internet Consumer Food safety and recalls Milk recalls
Food safety and recallsCurrently selected
Milk recalls
Lactalis Australia Pty Ltd has recently initiated two separate food recalls associated with milk products.
14 June 2019: Milk recall relating to chemical contamination
On 13 June 2019 Lactalis Australia Pty Ltd initiated a food recall to remove a number of Pauls and Coles branded milks from the marketplace, due to chemical contamination.
Find product details and date marks on our recall page.
The chemical is a food grade cleaning solution used for cleaning milk plant equipment. It poses a low public health risk and there have been no reports of illness. Affected milk products may have a yellowish colouring and/or metallic chemical taste.
The affected product had been for sale at Coles, Woolworths, IGAs, Independent stores and petrol stations in Victoria and in the Albury region (and surrounds) of NSW.
Lactalis Australia Pty Ltd has advised that all potentially contaminated product has now been recalled and the source of the problem has been identified. The company is working with the relevant authorities to ensure any issues have been addressed.
What should I do if I have purchased one of these affected products?
There is a very low risk to public health from this product. Consumers should dispose of the product or return it for a refund. If you have any health concerns seek medical advice.
FSANZ: Recall notice on FSANZ website
Coles recall: Recall of three varieties of Coles brand 1L fresh white milk
Woolworths: Recall Paul's Smarter White Milk 1L
20 June 2019: Milk recall relating to E.coli contamination
On 20 June 2019 Lactalis Australia Pty Ltd initiated a food recall to remove a number of different branded milks from the marketplace, due to Escherichia coli (E.coli) contamination.
The contamination was detected during routine company testing. The source of the problem was identified as a processing failure. Lactalis has advised that all potentially contaminated product has now been recalled, and all products manufactured at the site will only be released following microbiological testing to confirm product safety. The company is now working closely with relevant authorities to ensure product safety. There have been no reports of illness.
Consumers should dispose of the product or return it for a refund. If you have any health concerns seek medical advice. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,692 |
Outline of the Bank
List of Governors
Location (Guide Map to Head Office) / Visiting the Bank
Head Office, Branches, and Overseas Offices
Organization of the Bank
Policy Board
Responsibilities of the Governor, Deputy Governors, and Executive Directors
Organization of Head office, Branches and Offices
The Bank's Accounts
The Bank's Organizational Principles
Medium-Term Strategic Plan
Research Papers, Reports, Speeches and Statements Related to the Bank
Tours and Museums
Guides to the Bank
Exchange of Damaged Cash
Tours of the Bank's Head Office
Outline of Monetary Policy
Price Stability Target of 2 Percent and "Quantitative and Qualitative Monetary Easing with Yield Curve Control"
Monetary Policy Meetings
Summary of Opinions
Monetary Policy Releases
Monetary Policy Measures
Market Operations
Principal Terms and Conditions
Outlook for Economic Activity and Prices
Reports to the Diet
Research Papers, Reports, Speeches and Statements Related to Monetary Policy
Outline of Financial System Stability
Financial System Policy
On-Site Examination and Off-Site Monitoring
Fund Provision to Maintain an Orderly Financial System
Macroprudence
Center for Advanced Financial Technology
Interest Rate Benchmark Reform (Preparedness for the Discontinuation of LIBOR)
Special Deposit Facility to Enhance the Resilience of the Regional Financial System
Cooperation with the Financial Services Agency
Research Papers, Reports, Speeches and Statements Related to Financial System
Financial System Report
Other Releases Related to Financial System
Payments and Markets
Outline of Payment and Settlement Systems
Payment and Settlement Systems and the Bank
Payment and Settlement Systems Operated by the Private Sector
Operation of BOJ-NET
JGB Book-Entry System
FinTech Center
Cross-Industry Committee on Japanese Yen Interest Rate Benchmarks
Cross-Industry Forum on Interest Rate Benchmarks
Study Group of Market Participants
Repo Market Forum
Bond Market
Credit Market
Research Papers, Reports, Speeches and Statements Related to Payment and Markets
Payment and Settlement Systems Report
Market Operations in Each Fiscal Year
Market Functioning Survey concerning Climate Change
Other Releases Related to Payment and Markets
Banknotes, The Bank's Treasury Funds and JGS Services
General Information of Banknotes and Coins
A New Series of Banknotes and a New 500 Yen Coin
Banknotes and Coins in Use
Security Features of Bank of Japan Notes
Outline of Banknotes and Coins
Publications and Other Information
Treasury Funds Services
JGS Services
The Bank's Transactions with the Government
Research Papers, Reports, Speeches and Statements Related to Banknotes, Treasury Funds and JGS Services
Outline of International Finance
Foreign Currency Assets
Cooperation with Other Central Banks
Financial Cooperation in Asia
Research Papers, Reports, Speeches and Statements Related to International Finance
Other Releases Related to International Finance
Outline of Research and Studies
List of Reports & Research Papers
BOJ Reports & Research Papers
Regional Economic Report
Working Paper Series, Review Series, and Research Laboratory Series
Research Papers Released by IMES
Study Group Reports
Opinion Survey
Alternative Data Analysis
Discontinued Research Releases
Outline of Statistics and Statistical Release Schedule
Notices of Changes and Corrections
Explanations of Statistics
Notices of Changes and Revisions
Bank of Japan Statistics
Financial Institutions Accounts
Deposits and Loans Market
Flow of Funds
Related to BIS/FSB
Payment and Settlement
TANKAN
Balance of Payments Related Statistics
Discontinued Statistics / Revised Base Statistics
Data compiled in statistical publications
Historical Statistics on the Web Site of IMES
BOJ Time-Series Data Search
Social Networking Site Management Policy
About the Bank>
Speeches and Statements>
Speeches 2022>
Speech by Deputy Governor AMAMIYA at the Japan Society of Monetary Economics (Climate Change and Finance)
[Speech]Climate Change and FinanceSpeech at the Japan Society of Monetary Economics
AMAMIYA Masayoshi
Deputy Governor of the Bank of Japan
Full Text [PDF 691KB]
It gives me great pleasure to have this opportunity to speak to you today at the Japan Society of Monetary Economics.
Today, I would like to talk about climate change and finance. Addressing climate change has become one of the most important policy issues both internationally and at home. In corporate activities as well, addressing environmental, social, and governance (ESG) issues, including climate change, is an important management issue. The world of finance is also taking steps to address climate change. When making investments and loans, financial institutions increasingly consider whether they contribute to addressing climate change. In addition to shareholders, a wide range of stakeholders, including non-profit organizations, are demanding that financial institutions take climate change issues into account.
Climate change and finance have become increasingly linked in recent years in both policy and practice. In my speech today, I would like to first consider the basic issues related to climate change and finance, and then provide an outline of the efforts and initiatives of private financial institutions and central banks around the world, including the Bank of Japan. Finally, I would like to talk about some of the related challenges in financial and economic analysis, with a sincere request to all researchers for further research development.
I. Basic Issues Related to Climate Change and Finance
I would like to start by considering some basic issues related to climate change and finance. Please refer to Slide 1. One issue is why finance plays an important role in addressing climate change. Another issue is how climate change affects finance, in particular the financial system.
Importance of Finance in Addressing Climate Change
One way in which economics approaches climate change issues is through the concept of "externalities." In economics, when individual firms and households make economic decisions, they are assumed to do so based on the prices they face. However, the negative impact of associated greenhouse gas emissions on society is not necessarily reflected in those prices, resulting in excessive greenhouse gas emissions. To address these "externalities," an important role falls to fiscal policy, including the tax system such as carbon taxes and the provision of subsidies for investments that contribute to decarbonization, and regulations. However, to achieve net zero emissions, massive amounts of investment capital are required for business fixed investment to substantially increase the amount of electricity generated from renewable energy sources and for promoting innovation. The Glasgow Financial Alliance for Net Zero (GFANZ) has estimated that the world will need to invest $125 trillion to achieve net zero by 2050. In order to raise such huge amounts of investment capital, funding through financial institutions and financial markets has an important part to play in implementing investments to achieve net zero. As I mentioned earlier, not only are stakeholders such as shareholders demanding that financial institutions take climate change issues into account in their actions, but financial institutions themselves are also taking proactive steps, seeing the growing demand for funds to achieve net zero as a business opportunity. Moreover, for the vast amounts of funds to be used efficiently, it is essential that financial institutions and investors in financial markets choose investments carefully, appropriately assessing the risks and returns to ensure that funds are allocated efficiently. In addition, it is necessary to establish mechanisms that allow external stakeholders to closely monitor the appropriateness of the use of such funds. In this regard, an important issue is the enhancement of disclosures by firms and financial institutions. Enhanced disclosure helps to impose discipline on firms' and financial institutions' investment and financing activities through market pressure. Strengthening the voluntary efforts of firms and financial institutions through the discipline of disclosure represents a step toward "internalizing" the "externalities" of climate change into the market mechanism.
Impact of Climate Change on the Financial System
Next, I would like to consider how climate change will affect the financial system. In assessing the impact of climate change on the financial system, two types of risks need to be considered: physical risks and transition risks. "Physical risks" refer to risks that physical phenomena such as major disasters and rising sea levels caused by climate change will lead to losses for firms and households. "Transition risks" refer to risks that changes in policies, technologies, and consumer preferences associated with the transition to net zero will have an economic impact on firms and households. If the transition to decarbonization is delayed, "physical risks" will materialize, increasing the likelihood that natural disasters will trigger losses at financial institutions' borrowers. On the other hand, when the transition to net zero makes progress and financial institutions' existing borrowers are, for example, firms that emit a lot of carbon, the value of those borrower firms' assets will deteriorate. In both cases, the quantity and quality of financial institutions' investment and loans changes, which -- depending on the response -- could have a negative impact on the financial system. Therefore, addressing climate change issues has also become an important factor in terms of the management of financial institutions and the stability of the financial system.
II. Efforts by Private Financial Institutions
Next, I would like to discuss efforts by private financial institutions. As I mentioned earlier, private financial institutions have accelerated their efforts to address climate change in recent years. In this context, I would like to introduce two examples of cross-border cooperation and initiatives by financial institutions. Please refer to Slide 2. One example is GFANZ, a global voluntary alliance of private sector financial institutions that was formed last year. The alliance currently has more than 550 members, comprising a wide range of financial institutions, including banks, insurance companies, asset managers, institutional investors, etc., that discuss concrete measures in sector-specific alliances. Member financial institutions are required to develop individual plans toward net zero and publish relevant information. At last year's COP26, GFANZ members announced that a total of $100 trillion in investment and loans was committed to transforming the economy for net zero over the next three decades. As I mentioned, GFANZ has estimated that the world will need to invest approximately $125 trillion to achieve net zero by 2050. Therefore, it is important that financial institutions, including GFANZ members, hammer out active and concrete investment and lending policies in order to ensure that private financial institutions are able to fully finance the investment and loans needed to transition to decarbonization. In this context, there is growing recognition of the importance of transition financing. In June this year, GFANZ released a document on net-zero transition planning.1 In its recommendations and guidance for financial institutions, GFANZ has positioned a "managed phaseout of high emitting assets" as one of the pillars of the transition to net zero and emphasized the importance of transition financing.
Another effort by private financial institutions that has grown in influence in recent years is the Principles for Responsible Investment (PRI). This is an international group of institutional investors, formed at the initiative of the United Nations, that currently has about 5,000 participating institutions. Signatory institutions are required to take ESG factors into account in their investment decisions and report on their activities and progress toward implementing the principles. It is expected that institutional investors actively encourage firms that they invest in to move toward decarbonization.
1"Towards a Global Baseline for Net-zero Transition Planning," GFANZ, June 2022.
III. Initiatives by Central Banks
I would now like to discuss initiatives by central banks. Last year, the Bank of Japan established an internal organization, the "Climate Coordination Hub," to strengthen its climate change-related institutional setup, released the "The Bank of Japan's Strategy on Climate Change," and decided to implement a variety of measures.2
Initiatives are also underway at central banks overseas. The first to launch a comprehensive climate change initiative was the Bank of England. Similarly, the European Central Bank (ECB) decided an action plan last year. Meanwhile, the U.S. Federal Reserve established two committees last year to examine the implications of climate change for financial institutions and the financial system from a micro- and a macroprudential perspective, respectively.
Information sharing and cooperation among central banks on climate change issues is also making progress. The Network of Central Banks and Supervisors for Greening the Financial System (NGFS) was launched in 2017 as a forum for central banks and financial supervisors to discuss how to address climate change. The Bank of Japan joined in 2019, participating in discussions on the impact of climate change on the economy and the financial system. Based on the view that scenario analysis reflecting possible future climate conditions is useful for examining the risks to the financial system, the NGFS published a set of basic scenarios in 2020, which have subsequently been updated from time to time. Several central banks and financial supervisory authorities are currently conducting analyses of their financial system based on the NGFS scenarios. Climate change is also one of the central topics of discussion at the G7 and G20. Concretely, discussions include the promotion of corporate disclosures based on the TCFD recommendations, the design of disclosure standards by the International Sustainability Standards Board (ISSB), and the importance of scenario-based financial system analyses. The Executives' Meeting of East Asia-Pacific Central Banks (EMEAP), a gathering of Asian and Pacific central banks, has also discussed the impact of climate change. Asia accounts for about half of the world's carbon dioxide emissions and is also the region most vulnerable to flooding and other adverse effects of climate change. Asian policy makers therefore have strong interest in addressing climate change.
I would now like to turn to two topics -- central banks' mandate and market neutrality -- that are important issues for central banks in addressing climate change issues, and then discuss the Bank's specific initiatives, while also touching on the efforts of other central banks. Please refer to Slide 3.
2https://www.boj.or.jp/en/announcements/release_2021/rel210716b.htm/.
The Bank's website also provides a page with a list of links to information on the Bank's various activities with regard to climate change. See: https://www.boj.or.jp/en/about/climate/index.htm/.
Basic Concept 1: Central Banks' Mandate
I would like to start by considering central banks' mandate and the response to climate change. Central banks in many countries, including Japan, are responsible for price stability and the stability of the financial system. Since I already highlighted the importance of climate change issues to the financial system earlier, let me here talk about price stability. As I mentioned at the outset, increases in the frequency of disruptions to economic activities due to the loss of basic social infrastructure and in supply chain disruptions have been observed in recent years as a result of the growing scale and frequency of natural disasters worldwide due to the effects of climate change. These disruptions increase the volatility of real economic activity, which in turn leads to price fluctuations. In addition, there are concerns over swings in fossil fuel and other energy prices and their impact on other goods and services prices if the transition to net zero does not proceed smoothly. Therefore, if central banks' actions can help to smooth the transition to net zero, this will contribute to price stability in the medium to long term.
Thus, climate change may have an extremely large impact on the economy, prices, and financial conditions over the medium to long term. Therefore, from the standpoint of central banks, which are responsible for price stability and the stability of the financial system, supporting the private sector's response to climate change will contribute to macroeconomic stability in the long run.
Basic Concept 2: Market Neutrality
Next, I would like to talk about the response to climate change and market neutrality. Let me start with the basic concept of market neutrality. Central banks' actions affect society and the economy in many ways. The key principle in this regard is that the central bank should work on the macroeconomy as a whole while intervening as little as possible in the microeconomic allocation of resources. For example, central banks need to avoid making judgements about individual economic activities and financings to determine which are "brown" and which are "green." This raises the question in what form the central bank should take an active role in the response to climate change issues. If investment and lending decisions in the private sector were made without taking the negative externalities of greenhouse gases into account, market-neutral actions with respect to investments and loans currently implemented would preserve a resource allocation that is biased toward "brown" industries compared to what is socially desirable. On the other hand, if private-sector investments and loans were made taking the negative externalities of greenhouse gas emissions into account, central bank asset purchases and funds-supplying measures in line with those private-sector investment and loan portfolios would be more neutral toward the private-sector transition to net zero. Moreover, as mentioned earlier, firms and financial institutions are currently trying to internalize the negative externalities by becoming more proactive in addressing climate change. From a forward-looking perspective, supporting such developments would preserve market neutrality.
Based on these considerations, I would next like to outline the Bank's specific measures.
Last year, as one of its monetary policy measures, the Bank introduced a new fund-supplying measure, the so-called Climate Response Financing Operations, through which it provides funds to financial institutions for investments or loans that they make to address climate change based on their own decisions. Please see Slide 4. In principle, these Climate Response Financing Operations will be conducted twice a year, and funds were provided in December of last year and July of this year. Currently, 63 financial institutions, including regional financial institutions, are eligible for such funds, and the total outstanding balance of loans disbursed by the Bank is 3.6 trillion yen. Participating financial institutions are required to demonstrate their efforts to address climate change by disclosing information in the four thematic areas in the TCFD recommendations (governance, strategy, risk management, and metrics and targets) as well as targets and actual results for their investments or loans. On the other hand, to maintain market neutrality mentioned earlier and avoid involvement in the allocation of resources at the micro level, the Bank does not check individual investments and loans made by financial institutions. Thus, the setup of the Climate Response Financing Operations is that they support investments and loans to reduce carbon emissions through the provision of funds, while making use of the disclosure mechanisms as well as financial institutions' expertise in selecting investment and loans.
Meanwhile, the ECB has decided to change the way it purchases corporate bonds and treats collateral so that, based on certain rules, it takes into account the impact of climate change. In Europe, a taxonomy and corporate reporting standards on sustainability have been established, and the ECB's corporate bond purchases and treatment of collateral take these standards into account.
In Japan, on the other hand, the discussion on standards and taxonomies with regard to the response to climate change is still in flux. Therefore, under the Climate Response Financing Operations, it is financial institutions that make the decisions on which investments or loans contribute to addressing climate change, so that they can respond flexibly to firms' funding needs. In addition, the setup aims to impose market discipline by requiring financial institutions to disclose a certain level of information on their efforts to address climate change. Moreover, since financial intermediation in Japan is mainly based on indirect financing, the Climate Response Financing Operations, which provide funding for investments and loans that contribute to decarbonization via banks can be regarded as the most effective way to provide financial support not only to large corporations but also to small and medium-sized enterprises in their efforts to achieve net zero.
In the process of conducting the Climate Response Financing Operations, financial institutions have commented that the operations have provided them with an opportunity to enhance their own disclosure and that they have been able to enter a dialogue with their customers on business fixed investment to reduce carbon emissions. The Bank hopes that these operations will continue to work as a kind of catalyst and provide a push to efforts to reduce carbon emissions in Japan.
Next, I would like to talk about the response of the financial system. In pushing ahead with the transition to net zero, it is important that the proper functioning of financial intermediation is maintained. The Bank seeks to ensure the stability of Japan's financial system and the smooth functioning of financial intermediation by appropriately monitoring the situation and actively supporting financial institutions' efforts to identify and manage climate-related financial risks.
As mentioned earlier, climate change issues can have a significant impact on the business conditions of financial institutions, and hence on the stability of the financial system, through physical risks and transition risks. Please see Slide 5 for more details on the transmission channels. When physical risks and transition risks associated with climate change materialize, they can have a substantial impact on the real economy through the disruption of business, a decline in asset values, and a rise in energy prices. These adverse effects on the real economy will impact the financial system through a decline in the value of assets and an increase in credit costs.
Therefore, it is important for financial institutions to conduct appropriate risk management based on the impact of climate change. However, a key feature of climate change issues is that they are difficult to incorporate directly into existing risk management frameworks. The time horizon over which risks may materialize is extremely long and uncertain, and the data necessary for risk management do not exist. How to deal with these issues represents an urgent challenge not only for financial institutions but also for central banks and financial supervisory authorities.
Against this background, scenario analysis is currently attracting attention worldwide as one approach to quantitatively understand the impact on financial institutions and financial systems in the event that climate-related risks materialize. This approach involves assuming hypothetical scenarios that include greenhouse gas emissions, carbon prices, global average temperatures, as well as the corresponding paths of macro and industry-level economic variables, and then measuring the corresponding credit and market risks for each scenario. The scenarios are created using, at their core, the integrated assessment models developed by Professor Nordhaus of Yale University, winner of the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2018. Moreover, a number of risk models that have been used extensively in the field of finance are also employed to measure risk.
While scenario analysis employs essentially the same methodology as conventional stress tests conducted to check the resilience of financial institutions and the financial system to a substantial downturn in the economy, it differs in several respects. First, there are large differences in the availability of data and the econometric models. Conventional stress tests are based on data such as data of past business cycles and, building on previous analyses, can be performed with a reasonable degree of reliability. On the other hand, there is still insufficient data on climate change and related economic and financial variables, and there are no established integrated assessment models or risk measurement methods. Moreover, it has been pointed out that if climate change is not addressed, humanity will face unprecedented circumstances, and the negative economic impact may increase in a nonlinear fashion; however quantitative analysis of this is fraught with difficulties. The second difference concerns the time horizon of the risk analysis. While the time horizon for conventional stress testing is two to three years, assessing climate-related risks requires a quantitative understanding of the situation over a much longer horizon of several decades. Partly for these reasons, scenario analysis is still in its infancy.
Scenario analysis is now being undertaken in many countries. Please refer to Slide 6. According to a survey by the Financial Stability Board (FSB) and the NGFS, as of November 2022, about 40 countries or regions had already conducted or are planning to conduct scenario analysis. The objectives of scenario analysis include understanding the impact on individual financial institutions and the financial system as a whole, improving the analytical capacity of both authorities and financial institutions, and identifying data constraints. However, in most countries or regions, the use of scenario analysis is still at the trial stage.
In Japan, the Bank of Japan and the Financial Services Agency (FSA) jointly conducted a pilot scenario analysis exercise with major financial institutions last year, with the results released in August this year.3 The pilot exercise was not intended to provide a quantitative assessment of the impact of climate change on these major financial institutions. Rather, it was intended to serve as a means to continuously improve and develop analytical methods of the scenario analysis and focused on understanding data constraints, assessing the validity of analytical assumptions and methods, and identifying issues for future improvement and development. Specifically, the Bank and the FSA prepared scenarios based on those provided by the NGFS and let the major financial institutions conduct risk analyses of these scenarios using their own models. The results of the analysis showed that the major financial institutions had their own frameworks in place to perform risk analysis for various scenarios. On the other hand, they also showed that there was considerable variation in the risk measurement results due to differences in the data and measurement models used by the major financial institutions. This variation reflects the lack of data necessary for the analysis as well as the substantial uncertainty surrounding issues such as technological developments and client firms' behavior. Going forward, the Bank intends to further improve the sophistication of the scenario analysis by deepening its dialogue with financial institutions, taking the issues identified in the pilot exercise into account.
In addition, in its on-site examinations and offsite monitoring, the Bank makes efforts to continue its in-depth dialogue with financial institutions on their responses to climate-related financial risks and their support for client firms' efforts toward net zero. Moreover, the Bank also encourages financial institutions to enhance the quality and quantity of their disclosures based on the TCFD, etc., in light of the revised Corporate Governance Code.
3The results can be found here:
https://www.boj.or.jp/en/announcements/release_2022/rel220826a.htm/.
The "Market Functioning Survey Concerning Climate Change"
To address climate change issues, it is also important that financial markets fulfill their financial intermediation function. If risks and opportunities arising from climate change are appropriately reflected in the prices of financial instruments such as stocks and bonds, this should facilitate funding and investment through financial markets and, in turn, add impetus to the climate change response. Based on these considerations, the Bank decided to conduct a periodic survey of financial institutions, business corporations, rating agencies, and others to ascertain the functioning of climate change-related markets and the issues that need to be addressed in order to improve them. The results of the first survey were released in August.4 Please see Slide 7. The main findings were as follows. First, respondents indicated that although climate-related risks and opportunities were incorporated into the prices of financial instruments to some extent, there was still room for them to be incorporated to a greater extent. Second, with regard to the market for climate change-related ESG bonds, such as green bonds, the survey suggested that there was strong demand for ESG bonds. This appears to be due to the importance investors attach to making a social and environmental contribution. Third, in terms of issues in expanding the ESG bond market, in addition to the need to broaden the base of investors and issuers, some respondents pointed out issues regarding the availability of information and methods for assessing risks and opportunities. In October, the Bank held a conference call on the survey with about 150 participants from financial institutions, business corporations, rating agencies, industry associations, etc., to exchange opinions on the survey results and issues to be addressed to expand the ESG bond market.
Going forward, the Bank intends to continue conducting this survey on an annual basis, while making efforts to improve its contents, in order to understand the functioning of climate change-related markets and issues that need improvement, and to contribute to the development of these markets through dialogue with relevant stakeholders.
4The results can be found here: https://www.boj.or.jp/en/research/brp/ron_2022/ron220805a.htm/.
Central Banks' Own Operations
Next, I would like to talk about central banks' own operations, etc. There is a growing trend, particularly among European central banks, to conduct operations taking the need to respond to climate change into account. The Bank of Japan, as a business entity, is also taking steps to address climate change in its operations. Specifically, the Bank began disclosing information based on the TCFD's recommendations this year. In addition, under the existing principle that the Bank's foreign currency assets shall be managed with an emphasis on safety and liquidity, the Bank has been purchasing foreign currency-denominated green bonds issued by governments and other foreign institutions.
Furthermore, it is also important to participate and engage in international discussions on climate change issues, while closely monitoring developments in these discussions. At international gatherings such as the G7, G20, and EMEAP, as well as meetings with other central banks, the Bank collects information on efforts in other countries, explains the Bank's own measures, and contributes to progress in international efforts to address climate change by participating in multilateral discussions. With regard to the financial system, the Bank, in close collaboration with the FSA, has been actively involved in the development of an international framework for addressing climate-related financial risks at forums such as the Basel Committee on Banking Supervision, the FSB, the NGFS, and others. The Bank has also been cooperating with financial institutions and relevant authorities in international efforts to compile the data necessary to assess climate-related financial risks.
As part of its international financial cooperation, the Bank, in cooperation with other central banks, is strengthening its efforts to promote investment in climate-related financial products such as green bonds, with the aim of fostering the development of financial markets. The Bank has long been investing in the Asian Bond Fund launched by EMEAP for the purpose of supporting the development of local currency-denominated bond markets in Asia. In July last year, in consultation with other EMEAP member central banks, it was agreed to expand the scope of investment in the Asian Bond Fund in order to catalyze a further deepening of local-currency denominated green bond markets in the region, and this has been implemented starting March of this year.
IV. Challenges in the Financial and Economic Analysis of Climate-Related Issues
Next, I would like to discuss some of the challenges in the financial and economic analysis of climate change issues based on our experience to date. Please see Slide 8.
First, it is necessary to further develop analytical models. In particular, in order to refine scenario analyses to examine the economic impact of climate change on financial institutions and the financial system, it is essential to improve our theoretical analytical tools and accumulate empirical research. In addition, since climate change will have a significant impact on macroeconomic activity and prices, it may be necessary to take a different perspective in terms of theories of economic growth and the analysis of monetary policy. Last year, the Institute for Monetary and Economic Studies of the Bank of Japan published a series of special edition newsletters entitled "The Economics of Climate Change" to provide an easy-to-understand explanation of the current state of the economic analysis of climate change. Bank of Japan staff members have also deepened discussions with relevant experts, for example by actively participating in various study groups. I hope that the Bank will continue to share its research and practical expertise widely and carry on with the mutual exchange of ideas between the Bank and academia, and that academia continues to deepen its analyses of these issues, allowing the Bank to incorporate the insights obtained into its own work.
Second, the analysis of the financial and economic impact of climate change issues requires collaboration with experts in the natural sciences. This certainly applies to scenario analysis. For instance, the extent to which policy responses limit increases in global temperatures and hence the extent to which natural disasters are prevented is important information for measuring the economic impact of policy responses. The NGFS, which provides climate change scenarios to measure future economic risks, is working with climate scientists to improve the scenarios. Since climate change is both a global issue and at the same time something whose local impact varies considerably, there is a need for greater cooperation at the country or regional level between experts in the natural sciences and those examining the economic impact of climate change.
The third challenge concerns enhancements in terms of financial analysis. Various investments are needed to achieve net zero. As mentioned earlier, financing through financial institutions and financial markets is important to achieve such investments. A crucial aspect in this context is whether financial institutions' loan interest rates and the pricing of financial instruments adequately incorporate the risks and opportunities associated with dealing with climate change. The Market Functioning Survey conducted by the Bank of Japan mentioned earlier also indicated that there was room for risks and opportunities to be incorporated in such prices of financial instruments to a greater extent. Given the high uncertainty surrounding the risks and opportunities stemming from climate change, it may not be easy for markets to properly factor them in. Nevertheless, as a variety of analytical methods are tested and as data continues to accumulate, the extent to which efforts to address climate change issues are reflected in the prices of financial instruments is likely to increase, and this will support the smooth supply of investment funds needed to achieve net zero.
Fourth, there are challenges in terms of the data. For example, to capture the physical risks associated with flood damage, not only data on the flood damage itself but also granular data on the economic activities of households and firms, including geographical information, is needed. This would make it possible to quantify the flood damage risk to borrower firms and measure potential credit costs. So far, however, there is little granular data linking flood risk data and firms' activities. This gap between the data needed for analyses and the data actually available, i.e., the data gap, needs to be addressed.
The global pioneer in economic approaches to climate change is Professor Nordhaus of Yale University, whom I have mentioned earlier. Professor Nordhaus began examining the economic impact of global warming in the 1970s. As is well known, Professor Hirofumi Uzawa, who was my advisor when I was a student, also began his research on "social common capital" including the natural environment and the "social costs of automobiles" in the 1970s. This means that Japan also had a great pioneer in the economic analysis of climate change. I believe that it is our responsibility to follow Professor Uzawa's lead, to open up new frontiers in financial and economic analysis using currently available analytical tools and data, and to implement better policies based on the knowledge gained. I would like to conclude my speech by expressing my hope that members of the Japan Society of Monetary Economics will deepen their analysis of climate change issues and provide a variety of policy proposals.
2-1-1 Nihonbashi-Hongokucho,Chuo-ku,Tokyo (location)
Tel:+81-3-3279-1111
The Central Council for Financial Services Information
Institute for Monetary and Economic Studies
Copyright Bank of Japan All Rights Reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,822 |
\section{Introduction}
Dialogue systems are widely used in many real applications, i.e., mobile apps, virtual assistants, smart home and so on \cite{chen2017survey,DBLP:conf/acl/ZhanLLFWL20,DBLP:conf/emnlp/LiuZ0ZZ21,DBLP:conf/sigir/SongYBWWXWY21,DBLP:conf/sigir/XiaXY21}. Understanding user intent is a crucial step in dialogue systems \cite{DBLP:conf/naacl/YuHZDPL21,DBLP:conf/slt/0007KGLAG21,DBLP:conf/icassp/QinLCKZ021,DBLP:conf/sigir/YilmazT20,DBLP:conf/emnlp/QinCLWL19,DBLP:conf/acl/ShenHRJ20}, since its performance will directly affect the downstream decisions and policies. As user interests may change frequently over time and user expression is diverse, many new intents emerge quickly, which motivates zero-shot intent classification. However, related researches are still in infancy, only a few approaches are proposed to tackle this challenge.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\textwidth]{framework}
\caption{Illustration of the proposed framework. In the training procedure, utterances are first encoded by Bi-LSTM, and then extracted semantic features with the mixture attention mechanism. Specifically, the distributional signature attention utilizes all the seen data to learn the general word importance and leverages the label description information to estimate the class-specific word importance. Meanwhile, the MLP attention automatically calculates different weights for different words. Then the mixture attention, which combines the above two attentions, assigns more appropriate weights for different words. In the meta-learning paradigm, seen data is divided into meta-seen data for meta-training and meta-unseen data for meta-adapting. The model is first trained on meta-seen samples and then is fine-tuned on meta-unseen samples. In the testing procedure, utterances are first encoded by the mixture attention module, and then are classified by the final model which is obtained by the meta-learning paradigm.}
\label{framework}
\end{figure*}
Some works rely on some external resources like label ontologies, manually defined attributes or knowledge graph to find the relationship between seen and unseen intent labels \cite{ferreira2015zero,yazdani2015model,zhang2019integrating,DBLP:journals/corr/abs-2102-02925}. However, such resources are usually unavailable or difficult to obtain, this is because that collecting or producing these resources is labor intensive and time consuming.
To overcome the above issue, recent works utilize the word embeddings of intent labels instead of the external resources, which are easily to obtained via pretraining on text corpus. Specifically, \citet{baseline-cdssm} and \citet{baseline-zcnn} project the utterances and intent labels to a same semantic space, and then compute the similarities between utterances and intent labels. \citet{xia2018zero} leverage capsule networks \cite{DBLP:conf/nips/SabourFH17} to extract high-level semantic features, and transfer the prediction vectors from seen classes to unseen classes. \citet{liu2019reconstructing} introduce the dimensional attention mechanism to extract semantic features, and then reconstruct the transformation matrices in capsule networks for unseen intents by utilizing abundant latent information of the labeled utterances. \citet{DBLP:conf/acl/YanFLLZWL20} integrate the unknown intent identifier into the model \cite{liu2019reconstructing} to further improve the performance. \citet{DBLP:conf/acl/YeGCCXZWZC20} attempt to use the reinforced self-training framework to learn data selection strategy automatically. \citet{DBLP:conf/ijcai/SiL00LW21} propose a class-transductive framework which uses a multi-task learning objective function to find the inter-intent distinctions and a similarity scorer to associate the inter-intent connections. However, due to the instability in training and the imbalanced classification shift issue, i.e., the model tends to misclassify the unseen test instances into the seen classes, these methods still struggle in generalized zero-shot learning tasks.
After analyzing previous works comprehensively, we find that there are two key points for implementing zero-shot intent classification. The first is to extract better semantic features for utterances, and the second is to strengthen the model generalization ability. Inspired by the success of attention mechanism in representation learning \cite{DBLP:conf/nips/VaswaniSPUJGKP17} and meta-learning strategy in few-shot learning \cite{DBLP:conf/icml/FinnAL17}, we propose a simple yet effective meta-learning paradigm for zero-shot intent classification with mixture attention mechanism. Specifically, we first extract semantic features with the mixture attention mechanism, which can assign more reasonable weights for different words by leveraging the distributional signature attention and the multi-layer perceptron attention simultaneously, thus improving the quality of feature extraction greatly. Then to simulate the multiple zero-shot classification tasks on seen data, we divide seen data into meta-seen data for meta-training and meta-unseen data for meta-adapting, and train the model with a meta-training strategy, which helps to obtain better generalization ability.
The overall framework of our proposed model is shown in Figure~\ref{framework}. As far as we know, it is the first work to extend meta-learning to the \emph{Zero-Shot Intent Classification} (ZSIC) task. Extensive experimental results on two dialogue datasets in different languages demonstrate that the proposed model dramatically outperforms other baselines on both standard and generalized zero-shot intent classification tasks.
\section{The Proposed Method}
\subsection{Problem Formulation}
Zero-shot learning aims to predict the label $y^*$ of any test sample $\bm x^*$ belonging to an unseen class, leveraging the training data from seen classes. In general, there are two common settings. (1) Standard zero-shot classification: in this case $y^*\in\{C_{\text{unseen}}\}$; (2) Generalized zero-shot classification: in this case $y^*\in\{C_{\text{seen}}, C_{\text{unseen}}\}$. $C_{\text{seen}}$ and $C_{\text{unseen}}$ are the sets of seen and unseen intent classes respectively, and there is no overlap between them, i.e., $C_{\text{seen}} \bigcap C_{\text{unseen}} = \emptyset$.
\subsection{Extracting Semantic Features with Mixture Attention}
Given an utterance with $N$ words $[\bm w_{1},\bm w_{2},...,\bm w_{N}]$, where $\bm w_{t}\in \mathbb{R}^{d_{w} \times 1}$ is the $t$-th word embedding which can be pre-trained with any language model, we follow \cite{liu2019reconstructing,DBLP:conf/acl/YanFLLZWL20} to use a bidirectional LSTM \cite{DBLP:journals/neco/HochreiterS97} to pre-process these embeddings, i.e., $\bm h_{i}=\text{BiLSTM}(\bm w_{i}) \in \mathbb{R}^{2d_h \times 1}$. Then an utterance can be represented as a matrix $\bm H=[\bm h_1, \bm h_2, \ldots, \bm h_N] \in \mathbb{R}^{2d_h \times N}$, where $d_{h}$ is the hidden dimension of the forward/backward LSTM.
\subsubsection{Distributional Signature Attention}
Distributional signature attention has shown to be effective in few-shot text classification \cite{DBLP:conf/iclr/BaoWCB20}, which can utilize all the seen data to learn the general word importance and leverage the label description information to estimate the class-specific word importance.
\emph{(1) General word importance.} Frequently occurring words usually have limited discriminative information \cite{DBLP:journals/jd/Jones04}. To downweight frequent words and upweight rare words, we compute general word importance by $s(\bm w_t) = \frac{\varepsilon}{\varepsilon+\text{P}(\bm w_t)}$, where $\varepsilon = 10^{-5}$, and $\text{P}(\bm w_t)$ is unigram likelihood of the $t$-th word over all the seen data.
\emph{(2) Class-specific word importance.} As the importance of a word can be measured by its prediction result for different categories, we can calculate the class-specific word importance by $t(\bm w_{t}) = \mathcal{H}(\text{P}(y|\bm w_{t}))^{-1}$, where $ \mathcal{H}(\cdot)$ is the entropy operator, and $\text{P}(y|\bm w_{t})$ is the conditional probability of $y$ given $\bm w_t$, which can be obtained via a ridge regression classifier. $t(\bm w_t)$ measures the uncertainty of the class label $y$ given the word $\bm w_t$. So words with a skewed distribution will be highly weighted.
In zero-shot scenarios, the label description is the only known information of unseen data. We attempt to use it to calculate the class-specific word importance. Specifically, given the label description embedding matrix $\bm E =[\bm e_1, \bm e_2, \ldots, \bm e_C]^T \in \mathbb{R}^{C \times d_w}$, $\bm e_i$ is the embedding of the $i$-th label description which is computed by the average of its word embeddings, $\bm Y \in \mathbb{R}^{C \times C}$ is the one-hot label matrix, and $C$ is the total number of seen and unseen intent labels. Then we minimize the following regularized squared loss:
\begin{equation}
\mathcal{L} = \|\bm E \bm W-\bm Y\|^{2}_{F}+\beta\|\bm W\|^{2}_{F},
\end{equation}
where $\bm W \in \mathbb{R}^{d_w \times C}$ is the learnable weight matrix, $\|\cdot\|_F$ denotes the Frobenius norm. $\beta$ is the hyperparameter to avoid overfitting and we set $\beta =1$ consistently. Obviously, $\bm W$ has the closed-form solution:
\begin{equation}
\bm W = \bm E^T(\bm E \bm E^T+ \lambda \bm I)^{-1} \bm Y,
\end{equation}
where $\bm I$ is an identity matrix. Then given a word $\bm w_t \in \mathbb{R}^{d_w \times 1}$, we can estimate its conditional probability with:
\begin{equation}
\text{P}(y|\bm w_{t})=\text{softmax}(\bm w_t^T\bm W).
\end{equation}
\emph{(3) Distributional signature attention.} To combine the general and class-specific word importance, we use a bidirectional LSTM to deal with each word, i.e, $\bm z_{t}=\text{BiLSTM}(s(\bm w_{t})||t(\bm w_{t})) \in \mathbb{R}^{2d_b \times 1}$, where $d_b$ is the hidden dimension of the forward/backward LSTM. Then for an utterance, we can calculate the distributional signature attention $\bm p \in \mathbb{R}^{1 \times N}$ by:
\begin{equation}
\bm p = \text{softmax}(\bm F \bm Z),
\end{equation}
where $\bm Z=[\bm z_1, \bm z_2, \ldots, \bm z_N]\in \mathbb{R}^{2d_{b}\times N}$, and $\bm F \in \mathbb{R}^{1\times 2d_{b}}$ is a trainable parameter vector.
\subsubsection{Multi-layer Perceptron (MLP) Attention}
MLP attention can automatically calculate the appropriate weights for different words with the guide of loss function \cite{DBLP:conf/aaai/ShenZLJPZ18,DBLP:conf/acl/YanFLLZWL20}. Given an utterance representation $\bm H \in \mathbb{R}^{2d_h \times N}$, the MLP attention $\bm q \in \mathbb{R}^{1 \times N}$can be calculated with:
\begin{equation}
\bm q = \text{softmax}(\bm W_2(\text{ReLU}(\bm W_1 \bm H))),
\end{equation}
where $\bm W_1 \in \mathbb{R}^{d_{a}\times 2d_{h}}$ and $\bm W_2 \in \mathbb{R}^{1 \times d_{a}}$ are trainable parameter matrices.
\subsubsection{Mixture Attention}
Mixture attention aims to combine the distributional signature attention and MLP attention, and then assigns more appropriate weights for different words in an utterance. We use an adaptive weight vector $\bm b$ to integrate the distributional signature attention and the MLP attention. Specifically, the mixture attention $\bm a \in \mathbb{R}^{1 \times N}$ can be calculated with:
\begin{equation}
\bm a = \bm b [\bm p; \bm q],
\end{equation}
where $\bm p \in \mathbb{R}^{1 \times N}$ and $\bm q \in \mathbb{R}^{1 \times N}$ are the distributional signature attention and the MLP attention respectively, and $\bm b \in \mathbb{R}^{1\times 2}$ is a trainable parameter vector. Then given an utterance, the final semantic feature $\bm x \in \mathbb{R}^{2 d_h \times 1}$can be computed by:
\begin{equation}
\bm x = \bm H \bm a^T,
\end{equation}
where $\bm a^T$ is the transpose of $\bm a$. Hereinafter, we use $\bm x$ to represent an utterance or an utterance embedding.
\subsection{Meta-learning Paradigm for ZSIC}
Meta-learning has shown promising performance in few-shot learning \cite{DBLP:conf/nips/JeongK20,DBLP:conf/nips/PatacchiolaTCOS20,DBLP:conf/ijcai/SunOZD21,DBLP:conf/ijcai/ZhuLJ20,DBLP:conf/acl/HanFZQGZ21}, while only a few meta-learning based methods are designed for zero-shot learning, which leverage some complex models like VAE or GAN \cite{yu2020episode,verma2020meta}. In addition, these methods mainly concentrate on image domain, and are difficult to directly applied in the intent classification task.
In this work, we propose a simple meta-learning paradigm for \emph{Zero-Shot Intent Classification} (ZSIC), which is trained with the episode strategy. Different from existing meta-learning based few-shot approaches, each episode in our method mimics a zero-shot classification task. In particular, we divide each episode into two phases: \emph{Meta-training} and \emph{Meta-adapting}. Accordingly, to simulate a fake ZSIC task, we randomly divide the seen classes $C_{seen}$ into the meta-seen classes $C_{\text{meta-s}}$ (the samples belonging to $C_{\text{meta-s}}$ are denoted by $X_{\text{meta-s}}$ and used in meta-training phase) and the meta-unseen classes $C_{\text{meta-u}}$ (the samples belonging $C_{\text{meta-u}}$ are denoted by $X_{\text{meta-u}}$ and used in meta-adapting phase). Note that $C_{\text{meta-u}}\cap C_{\text{meta-s}}=\varnothing$.
\subsubsection{Meta-training} Intuitively, the class label description usually contains lots of class-indicative information, so it can be treated as the class prototype (center). If we can learn a model $G$ to project the class label description to the utterance representation space, the embeddings of utterances belonging to the same class should be close to the projection embedding of their label description.
Assume $\bm e_{i} \in \mathbb{R}^{d_w \times 1}$ is the embedding of the $i$-th label description in meta-seen classes, and the model $G$ is a two-layer neural network, we can project $\bm e_i$ to the utterance representation space by:
\begin{equation}\label{LtoU}
G(\bm e_{i})=\text{Tanh}(\bm M_{2}\text{Tanh}(\bm M_{1}\bm e_{i})),
\end{equation}
where $G(\bm e_i) \in \mathbb{R}^{2d_h \times 1}$ denotes the class prototype of the $i$-th category in the utterance representation space. $\bm M_{1} \in \mathbb{R}^{d_s\times d_w}$ and $\bm M_{2} \in \mathbb{R}^{2d_h\times d_s}$ are trainable parameter matrices. Then the probability that an utterance $\bm x$ belongs to the $i$-th category can be calculated by:
\begin{equation}
p_{i}(\bm x)=\frac{exp(-d(\bm x,G(\bm e_{i})))}{\sum_{j}exp(-d(\bm x,G(\bm e_{j})))},
\end{equation}
where $d$ denotes the Euclidean distance.
In the meta-training procedure, we aim to maximize the probabilities between each utterance and its corresponding class prototype, which can be reformulated by minimizing the following loss function:
\begin{equation}
\mathcal{L}_{train} = -\sum_{\bm x \in X_{\text{meta-s}}}logp_{i}(\bm x).
\end{equation}
By optimizing $\mathcal{L}_{train}$ with gradient descent, each utterance will be forced to have higher affinity with its corresponding class prototype. During meta-training, we take all utterances in meta-seen classes to train all parameters in the whole model, including mixture attention module and meta-learning module.
\subsubsection{Meta-adapting} To enhance the model generalization ability, we add the meta-adapting phase to refine the model parameters in $G$ using meta-unseen classes. Specifically, given an utterance $\bm x \in X_{\text{meta-u}}$, we use the model learned in meta-training phase to obtain its probability distribution over different classes, i.e., $p(y=i| \bm x)= \frac{exp(-d(\bm x,G(\bm e_{i})))}{\sum_{j}exp(-d(\bm x,G(\bm e_{j})))}$, where $\bm e_{i}$ is the embedding of the $i$-th label description in meta-unseen classes. By minimizing the negative log-probability of the true class $y_k$, i.e.,
\begin{equation}
\mathcal{L}_{adapt}=-\sum_{\bm x \in X_{\text{meta-u}}}log p(y=k|\bm x),
\end{equation}
the model parameters in $G$ can be further updated to adapt the unseen classes.
\subsubsection{Testing} Given a test utterance $\bm x^*$, its class label $y^*$ can be predicted by:
\begin{equation}
y^*=\arg \min_k(d(\bm x^*,G(\bm e_{k})),
\end{equation}
where $d$ denotes the Euclidean distance. $\bm e_{k}$ is the embedding of the $k$-th label description. $G(\bm e_{k})$ is the class prototype of the $k$-th category in the utterance representation space.
\section{Experiments}
\begin{table}[t]
\centering
\caption{Dataset statistics.}
\begin{tabular}{l|c|c}
\toprule
\textbf{Dataset} & \textbf{SNIPS} & \textbf{SMP} \\
\midrule
Vocab Size & 11641 & 2682 \\
Number of Samples & 13802 & 2460 \\
Average Sentence Length & 9.05 & 4.86 \\
Number of Seen Intents & 5 & 24 \\
Number of Unseen Intents & 2 & 6 \\
\bottomrule
\end{tabular}
\label{tab:dataset}
\end{table}
\begin{table*}[t]
\small
\centering
\caption{Results of generalized zero-shot classification. `Seen', `Unseen' and `Overall' denote the performance on the utterances from seen intents, unseen intents, and both seen and unseen intents respectively.}
\tabcolsep=0.17cm
\begin{tabular}{l|cc|cc|cc|cc|cc|cc}
\toprule
{} & \multicolumn{6}{c|}{\textbf{SNIPS}} & \multicolumn{6}{c}{\textbf{SMP}}\\
\cmidrule(lr){2-13}
{\textbf{Method}} & \multicolumn{2}{c|}{\textbf{Seen}} & \multicolumn{2}{c|}{\textbf{Unseen}} & \multicolumn{2}{c|}{\textbf{Overall}} & \multicolumn{2}{c|}{\textbf{Seen}} & \multicolumn{2}{c|}{\textbf{Unseen}} & \multicolumn{2}{c}{\textbf{Overall}}\\
\cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9} \cmidrule(lr){10-11} \cmidrule(lr){12-13}
{} & Acc & F1 & Acc & F1& Acc & F1 & Acc & F1 & Acc & F1& Acc & F1\\
\midrule
DeViSE \cite{baseline-devise} & 0.9481 & 0.6536 & 0.0211 & 0.0398 & 0.4215 & 0.3049 & 0.8040 & 0.6740 & 0.0270 & 0.0310 & 0.5030 & 0.4250 \\
CMT \cite{baseline-cmt} & \textbf{0.9755} & 0.6648 & 0.0397 & 0.0704 & 0.4438 & 0.3271 & 0.8314 & 0.7221 & 0.0798 & 0.1069 & 0.5398 & 0.4834 \\
CDSSM \cite{baseline-cdssm} & 0.9549 & 0.7033 & 0.0111 & 0.0218 & 0.4234 & 0.3194 & 0.6653 & 0.5540 & 0.1436 & 0.1200 & 0.4864 & 0.4052 \\
ZSDNN \cite{baseline-zcnn} & 0.9432 & 0.6679 & 0.0682 & 0.1041 & 0.4488 & 0.3493 & 0.7323 & 0.6116 & 0.0590 & 0.0869 & 0.5013 & 0.4316 \\
IntentCapsNet \cite{xia2018zero} & 0.9741 & 0.6517 & 0.0000 & 0.0000 & 0.4200 & 0.2810 & \textbf{0.8850} & 0.7281 & 0.0000 & 0.0000 & 0.5375 & 0.4423 \\
ReCapsNet \cite{liu2019reconstructing} & 0.9664 & 0.6743 & 0.1121 & 0.1764 & 0.4805 & 0.3911 & 0.8230 & 0.7450 & 0.1720 & 0.1526 & 0.5674 & 0.5124 \\
RL Self-training \cite{DBLP:conf/acl/YeGCCXZWZC20} & 0.7391 & 0.7558 & 0.5505 & 0.6901 & 0.6257 & 0.7182 & 0.5254 & 0.4791 & 0.4538 & 0.4479 & 0.4654 & 0.4868 \\
CTIR \cite{DBLP:conf/ijcai/SiL00LW21}
& 0.9693 & 0.6524 & 0.0067 & 0.0132 & 0.4220 & 0.2890 & 0.8322 & 0.7016 & 0.1839 & 0.2083 & 0.5774 & 0.5077 \\
SEG \cite{DBLP:conf/acl/YanFLLZWL20} & 0.8644 & \textbf{0.8658} & 0.6961 & 0.6931 & 0.7685 & 0.7674 & 0.6821 & 0.7359 & 0.4848 & 0.3806 & 0.6046 & 0.5963 \\
\midrule
Ours (w/o gw) & 0.6328 & 0.5236 & 0.4291 & 0.4671 & 0.5170 & 0.4915 & 0.7584 & 0.7320 & 0.3964 & 0.3942 & 0.6161 & 0.5992 \\
Ours (w/o cw) & 0.7514 & 0.6143 & 0.4714 & 0.5512 & 0.5922 & 0.5784 & 0.7383 & 0.7179 & 0.3808 & 0.3051 & 0.5978 & 0.5556\\
Ours (w/o DS attention) & 0.7537 & 0.6036 & 0.4671 & 0.5342 & 0.5907 & 0.5641 & 0.7970 & \textbf{0.7464} & 0.2772 & 0.2868 & 0.5927 & 0.5657 \\
Ours (w/o MLP attention) & 0.7551 & 0.5944 & 0.4835 & 0.5566 & 0.6006 & 0.5729 & 0.8087 & 0.7446 & 0.2047 & 0.2268 & 0.5713 & 0.5411 \\
Ours (w/o meta-adapting) & 0.9507 & 0.6702 & 0.2071 & 0.3267 & 0.5279 & 0.4749 & 0.7651 & 0.7159 & 0.2876 & 0.2916 & 0.5774 & 0.5491 \\
\midrule
Ours & 0.7588 & 0.5944 & 0.5053 & 0.5576 & 0.6146 & 0.5735 & 0.7466 & 0.7268 & 0.4352 & 0.4107 & 0.6242 & 0.6026\\
Ours (SEG) & 0.7750 & 0.7903 & \textbf{0.7708} & \textbf{0.7597} & \textbf{0.7726} & \textbf{0.7729} & 0.7061 & 0.7046 & \textbf{0.5025} & \textbf{0.4740} & \textbf{0.6261} & \textbf{0.6140} \\
\bottomrule
\end{tabular}
\label{tab:result1}
\end{table*}
\subsection{Datasets and Splitting}
\subsubsection{Datasets} We follow \cite{liu2019reconstructing,DBLP:conf/acl/YanFLLZWL20} to evaluate our method on two real dialogue datasets: SNIPS \cite{data_snip} and SMP \cite{data_smp}. SNIPS is an open-source single turn English corpus dialogue, which includes crowdsourced queries distributed among 7 user intents. SMP is a Chinese dialogue corpus for user intent classification released in China National Conference on Social Media Processing. The detailed dataset statistics are summarized in Table \ref{tab:dataset}.
\subsubsection{Data Splitting} For data splitting, we follow \cite{liu2019reconstructing,DBLP:conf/acl/YanFLLZWL20} to divide the datasets. Specifically, for standard zero-shot intent classification, we take all the samples of seen intents as the training set, and all the samples of unseen intents as the test set. For generalized zero-shot intent classification, we randomly take 70\% samples of each seen intent as the training set, and the remaining 30\% samples of each seen intent and all the samples of unseen intents as the test set.
\subsection{Baselines}
We compare the proposed model with the following state-of-the-art baselines: DeViSE \cite{baseline-devise}, CMT \cite{baseline-cmt}, CDSSM \cite{baseline-cdssm}, ZSDNN \cite{baseline-zcnn}, IntentCapsNet \cite{xia2018zero}, ReCapsNet \cite{liu2019reconstructing}, RL Self-training\cite{DBLP:conf/acl/YeGCCXZWZC20}, CTIR \cite{DBLP:conf/ijcai/SiL00LW21} and SEG \cite{DBLP:conf/acl/YanFLLZWL20}. SEG is a plug-and-play unknown intent detection method, and it integrates with ReCapsNet in the original paper. For fair comparison, we combine our method with SEG and test it in generalized ZSIC. Note that SEG is unsuitable in standard ZSIC as all test utterances are from unseen classes.
In addition, we conduct ablation study to evaluate the contributions of different components in our model. Specifically, we test our model without (w/o) general word importance (gw), class-specific word importance (cw), distributional signature (DS) attention, MLP attention and meta-adapting phase respectively.
\subsection{Implementation Details}
\subsubsection{Evaluation Metrics}
We adopt two widely used evaluation metrics: accuracy (ACC) and micro-averaged F1 scores (F1) to evaluate the performance. Both metrics are computed with the average value weighted by the support of each class, where the support means the sample ratio of the corresponding class.
\subsubsection{Experiment Settings}
In terms of embeddings, for SNIPS, we use the embeddings pre-trained on English Wikipedia \cite{emb_snip}. For SMP, we use the Chinese word embeddings pre-trained by \cite{emb_smp}. We also evaluate our model with word embeddings pre-trained by BERT \cite{naacl/DevlinCLT19} on the standard zero-shot classification task. In generalized zero-shot tasks, to alleviate the issue that the model tends to classify samples into seen classes, we set a threshold $\lambda$. If a sample's maximum predictive value is less than $\lambda$, the model will classify it among unseen classes. And we set $\lambda=0.6$ on SNIPS, and $\lambda=0.8$ on SMP respectively.
In addition, we use the Adam optimizer \cite{DBLP:journals/corr/KingmaB14} to train the proposed model. For SNIPS, in standard and generalized zero-shot intent classification, we set the initial learning rate 0.006 and 0.002 for meta-training and meta-adapting respectively. And in each episode, we randomly select 4 intents from seen intents as meta-seen classes, and the remaining intents as meta-unseen classes. For SMP, in standard and generalized zero-shot intent classification, we set the initial learning rate 0.008 and 0.004 for meta-training and meta-adapting respectively. And in each episode, we randomly select 21 intents from seen intents as meta-seen classes, and the remaining intents as meta-unseen classes.
\begin{table}[t]
\small
\caption{Results of standard zero-shot classification.}
\centering
\tabcolsep=0.17cm
\begin{tabular}{l|cc|cc}
\toprule
{\textbf{Method}} & \multicolumn{2}{c|}{\textbf{SNIPS}} & \multicolumn{2}{c}{\textbf{SMP}}\\
\cmidrule(lr){2-5}
{} & Acc & F1 & Acc & F1\\
\midrule
DeViSE \cite{baseline-devise} & 0.7447 & 0.7446 & 0.5456 & 0.3875 \\
CMT \cite{baseline-cmt} & 0.7396 & 0.7206 & 0.4452 & 0.4245 \\
CDSSM \cite{baseline-cdssm} & 0.7588 & 0.7580 & 0.4308 & 0.3765 \\
ZSDNN \cite{baseline-zcnn} & 0.7165 & 0.7116 & 0.4615 & 0.3897 \\
IntentCapsNet \cite{xia2018zero} & 0.7752 & 0.7750 & 0.4864 & 0.4227 \\
ReCapsNet \cite{liu2019reconstructing} & 0.7996 & 0.7980 & 0.5418 & 0.4769 \\
RL Self-training \cite{DBLP:conf/acl/YeGCCXZWZC20} & 0.8253 & 0.8726 & 0.7124 & 0.6587 \\
CTIR \cite{DBLP:conf/ijcai/SiL00LW21} & 0.6865 & 0.6823 & 0.5462 & 0.5739 \\
\midrule
Ours (w/o gw) & 0.6832 & 0.6676 & 0.5440 & 0.5492 \\
Ours (w/o cw) & 0.7349 & 0.7342 & 0.5233 & 0.4460 \\
Ours (w/o DS attention) & 0.7480 & 0.7471 & 0.5492 & 0.5292 \\
Ours (w/o MLP attention) & 0.8111 & 0.8072 & 0.4741 & 0.4175 \\
Ours (w/o meta-adapting) & 0.8390 & 0.8370 & 0.5751 & 0.5436\\
\midrule
Ours & 0.8444 & 0.8443 & 0.6088 & 0.5803 \\
Ours (BERT) & \textbf{0.9139} & \textbf{0.9135} & \textbf{0.7335} & \textbf{0.7022} \\
\bottomrule
\end{tabular}
\label{tab:result2}
\end{table}
\subsection{Result Analysis}
Table \ref{tab:result1} and Table \ref{tab:result2} report the experimental results of generalized zero-shot classification and standard zero-shot classification respectively. Some baselines results are taken from \cite{DBLP:conf/acl/YanFLLZWL20}. And the best result is highlighted in bold. Based on the results, we can make the following observations.
\begin{itemize}
\item In the standard zero-shot intent classification task, our model outperforms other strong baselines on both SNIPS and SMP, which validates the effectiveness of our model in dealing with zero-shot intent classification. In addition, we can also observe that the pre-trained model BERT can improve our model performance effectively.
\item In the generalized zero-shot intent classification task, our model performs worse than some methods in detecting seen intents. This is because that some baselines tend to classify the test utterances as seen intents, which also explains the reason that these methods perform much worse in detecting unseen intents. From the overall performance, it can be observed that our method can achieve satisfactory results in most cases.
\item When combining with SEG, our method performs much better than the original version, which indicates that unknown intent detection is helpful to improve generalized zero-shot intent classification and achieve remarkable performance.
\item From ablation study, it can be seen that all modules contribute to the model to some extent, which proves that the mixture attention is powerful and the meta-adapting phase is indispensable.
\end{itemize}
\section{Conclusion}
In this paper, we propose a novel meta-learning paradigm for zero-shot intent classification. The performance gains of the proposed method come from two aspects. By constructing mixture attention mechanism, more reasonable word importance can be learned to obtain better semantic features for utterances. By introducing meta-learning paradigm, the model can achieve better generalization ability to classify utterances in unseen classes. Extensive experimental results confirm the superiority of our method on both standard and generalized zero-shot intent classification tasks. In future work, we plan to extend our framework to deal with multiple-label zero-shot intent detection, and explore more meta-learning strategies for zero-shot intent classification.
\begin{acks}
The authors are grateful to the anonymous reviewers for their valuable comments and suggestions. This work was supported by National Natural Science Foundation of China (No. 62106035, 61876028), and Fundamental Research Funds for the Central Universities (No. DUT20RC(3)040, No. DUT20RC(3)066).
\end{acks}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,168 |
package org.eclipse.jdt.internal.ui.workingsets;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Iterator;
import java.util.List;
import org.eclipse.core.runtime.IAdaptable;
import org.eclipse.core.resources.IProject;
import org.eclipse.core.resources.IResource;
import org.eclipse.core.resources.IResourceDelta;
import org.eclipse.core.resources.ResourcesPlugin;
import org.eclipse.ui.IWorkingSet;
import org.eclipse.ui.IWorkingSetUpdater;
import org.eclipse.jdt.core.ElementChangedEvent;
import org.eclipse.jdt.core.IElementChangedListener;
import org.eclipse.jdt.core.IJavaElement;
import org.eclipse.jdt.core.IJavaElementDelta;
import org.eclipse.jdt.core.IJavaProject;
import org.eclipse.jdt.core.JavaCore;
public class JavaWorkingSetUpdater implements IWorkingSetUpdater, IElementChangedListener {
/**
* DO NOT REMOVE, used in a product, see https://bugs.eclipse.org/297529 .
* @deprecated As of 3.5, replaced by {@link IWorkingSetIDs#JAVA}
*/
public static final String ID= IWorkingSetIDs.JAVA;
private List<IWorkingSet> fWorkingSets;
private static class WorkingSetDelta {
private IWorkingSet fWorkingSet;
private List<IAdaptable> fElements;
private boolean fChanged;
public WorkingSetDelta(IWorkingSet workingSet) {
fWorkingSet= workingSet;
fElements= new ArrayList<IAdaptable>(Arrays.asList(workingSet.getElements()));
}
public int indexOf(Object element) {
return fElements.indexOf(element);
}
public void set(int index, IAdaptable element) {
fElements.set(index, element);
fChanged= true;
}
public void remove(int index) {
if (fElements.remove(index) != null) {
fChanged= true;
}
}
public void process() {
if (fChanged) {
fWorkingSet.setElements(fElements.toArray(new IAdaptable[fElements.size()]));
}
}
}
public JavaWorkingSetUpdater() {
fWorkingSets= new ArrayList<IWorkingSet>();
JavaCore.addElementChangedListener(this);
}
/**
* {@inheritDoc}
*/
public void add(IWorkingSet workingSet) {
checkElementExistence(workingSet);
synchronized (fWorkingSets) {
fWorkingSets.add(workingSet);
}
}
/**
* {@inheritDoc}
*/
public boolean remove(IWorkingSet workingSet) {
boolean result;
synchronized(fWorkingSets) {
result= fWorkingSets.remove(workingSet);
}
return result;
}
/**
* {@inheritDoc}
*/
public boolean contains(IWorkingSet workingSet) {
synchronized(fWorkingSets) {
return fWorkingSets.contains(workingSet);
}
}
/**
* {@inheritDoc}
*/
public void dispose() {
synchronized(fWorkingSets) {
fWorkingSets.clear();
}
JavaCore.removeElementChangedListener(this);
}
/**
* {@inheritDoc}
*/
public void elementChanged(ElementChangedEvent event) {
IWorkingSet[] workingSets;
synchronized(fWorkingSets) {
workingSets= fWorkingSets.toArray(new IWorkingSet[fWorkingSets.size()]);
}
for (int w= 0; w < workingSets.length; w++) {
WorkingSetDelta workingSetDelta= new WorkingSetDelta(workingSets[w]);
processJavaDelta(workingSetDelta, event.getDelta());
IResourceDelta[] resourceDeltas= event.getDelta().getResourceDeltas();
if (resourceDeltas != null) {
for (int r= 0; r < resourceDeltas.length; r++) {
processResourceDelta(workingSetDelta, resourceDeltas[r]);
}
}
workingSetDelta.process();
}
}
private void processJavaDelta(WorkingSetDelta result, IJavaElementDelta delta) {
IJavaElement jElement= delta.getElement();
int index= result.indexOf(jElement);
int type= jElement.getElementType();
int kind= delta.getKind();
int flags= delta.getFlags();
if (type == IJavaElement.JAVA_PROJECT && kind == IJavaElementDelta.CHANGED) {
if (index != -1 && (flags & IJavaElementDelta.F_CLOSED) != 0) {
result.set(index, ((IJavaProject)jElement).getProject());
} else if ((flags & IJavaElementDelta.F_OPENED) != 0) {
index= result.indexOf(((IJavaProject)jElement).getProject());
if (index != -1)
result.set(index, jElement);
}
}
if (index != -1) {
if (kind == IJavaElementDelta.REMOVED) {
if ((flags & IJavaElementDelta.F_MOVED_TO) != 0) {
result.set(index, delta.getMovedToElement());
} else {
result.remove(index);
}
}
}
IResourceDelta[] resourceDeltas= delta.getResourceDeltas();
if (resourceDeltas != null) {
for (int i= 0; i < resourceDeltas.length; i++) {
processResourceDelta(result, resourceDeltas[i]);
}
}
IJavaElementDelta[] children= delta.getAffectedChildren();
for (int i= 0; i < children.length; i++) {
processJavaDelta(result, children[i]);
}
}
private void processResourceDelta(WorkingSetDelta result, IResourceDelta delta) {
IResource resource= delta.getResource();
int type= resource.getType();
int index= result.indexOf(resource);
int kind= delta.getKind();
int flags= delta.getFlags();
if (kind == IResourceDelta.CHANGED && type == IResource.PROJECT && index != -1) {
if ((flags & IResourceDelta.OPEN) != 0) {
result.set(index, resource);
}
}
if (index != -1 && kind == IResourceDelta.REMOVED) {
if ((flags & IResourceDelta.MOVED_TO) != 0) {
result.set(index,
ResourcesPlugin.getWorkspace().getRoot().findMember(delta.getMovedToPath()));
} else {
result.remove(index);
}
}
// Don't dive into closed or opened projects
if (projectGotClosedOrOpened(resource, kind, flags))
return;
IResourceDelta[] children= delta.getAffectedChildren();
for (int i= 0; i < children.length; i++) {
processResourceDelta(result, children[i]);
}
}
private boolean projectGotClosedOrOpened(IResource resource, int kind, int flags) {
return resource.getType() == IResource.PROJECT
&& kind == IResourceDelta.CHANGED
&& (flags & IResourceDelta.OPEN) != 0;
}
private void checkElementExistence(IWorkingSet workingSet) {
List<IAdaptable> elements= new ArrayList<IAdaptable>(Arrays.asList(workingSet.getElements()));
boolean changed= false;
for (Iterator<IAdaptable> iter= elements.iterator(); iter.hasNext();) {
IAdaptable element= iter.next();
boolean remove= false;
if (element instanceof IJavaElement) {
IJavaElement jElement= (IJavaElement)element;
// If we have directly a project then remove it when it
// doesn't exist anymore. However if we have a sub element
// under a project only remove the element if the parent
// project is open. Otherwise we would remove all elements
// in closed projects.
if (jElement instanceof IJavaProject) {
remove= !jElement.exists();
} else {
final IJavaProject javaProject= jElement.getJavaProject();
final boolean isProjectOpen= javaProject != null ? javaProject.getProject().isOpen() : true;
remove= isProjectOpen && !jElement.exists();
}
} else if (element instanceof IResource) {
IResource resource= (IResource)element;
// See comments above
if (resource instanceof IProject) {
remove= !resource.exists();
} else {
IProject project= resource.getProject();
remove= (project != null ? project.isOpen() : true) && !resource.exists();
}
}
if (remove) {
iter.remove();
changed= true;
}
}
if (changed) {
workingSet.setElements(elements.toArray(new IAdaptable[elements.size()]));
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,888 |
Mosannona discolor är en kirimojaväxtart som först beskrevs av Robert Elias Fries, och fick sitt nu gällande namn av Laurentius 'Lars' Willem Chatrou. Mosannona discolor ingår i släktet Mosannona och familjen kirimojaväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Kirimojaväxter
discolor | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,902 |
Q: Set axis depending draggable direction jQuery draggable How can the axis be set depending it's draggable direction (after init)?
If it's dragged left-right it will be x, y if dragged up-down.
$("#belt").draggable({
handle: "li",
axis: "y",
start: function() {
//I want to be dragged in the axis i belong which should be x....
},
});
A: You use distance to constrain the motion initially to get the initial read of which direction the user is moving and then set the axis limitation.
var x, y;
$("#belt").draggable({
start: function(event) {
x = event.originalEvent.pageX;
y = event.originalEvent.pageY;
},
drag: function(event) {
if (x && y) {
axis = Math.abs(event.originalEvent.pageX - x) > Math.abs(event.originalEvent.pageY - y) ? 'x' : 'y';
$("#belt").draggable('option', 'axis', axis);
x = y = null;
}
},
stop: function() {
x = y = null;
$("#belt").draggable('option', 'axis', false);
},
distance: 20
});
Here's a fiddle: http://jsfiddle.net/jhchen/B7J2E/
A: Appreciate the answer jhchen! I forked your fiddle to include some code so that the axis locking happens only when the user presses and holds the shift key while dragging.
http://jsfiddle.net/B7J2E/40/
var x, y;
$("#test").draggable({
start: function(event) {
console.log(event);
x = event.originalEvent.pageX;
y = event.originalEvent.pageY;
console.log(x, y);
},
drag: function(event) {
if (event.shiftKey) {
if (x && y) {
axis = Math.abs(event.originalEvent.pageX - x) > Math.abs(event.originalEvent.pageY - y) ? 'x' : 'y';
$("#test").draggable('option', 'axis', axis);
x = y = null;
}
}
},
stop: function() {
x = y = null;
$("#test").draggable('option', 'axis', false);
},
distance: 20
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,626 |
\section*{ }
The spectrum of the scalar meson nonet is a long-standing problem of light
meson spectroscopy. The number of resonances found in
the region of 1--2 GeV exceeds the number of states that conventional quark
models can accommodate \cite{pdg}. Extra states are interpreted alternatively
as $K\bar{K}$ molecules, glueballs, multi-quark states or hybrids. In
particular, except for a well established scalar isodoublet state, the $K_0^
\ast (1430),$ the Particle Data Group (PDG) \cite{pdg} lists two isovector
states, the $a_0(980)$ and $a_0(1450).$ The latter, having mass and width
1450$\pm 40$ MeV, $270\pm 40$ MeV, respectively, was discovered recently by
the Crystal Barrel collaboration \cite{LEAR}. A third isovector state (not
included in \cite{pdg}), $a_0(1320),$ having mass and width $1322\pm 30$ MeV
and $130\pm 30$ MeV, was seen by GAMS \cite{GAMS} and LASS \cite{Aston} in the
partial wave analyses of the $\eta \pi $ and $K_sK_s$ data, respectively.
There are four isoscalar states in \cite{pdg}, the $f_0(400-1200)$ (or $\sigma
),$ the interpretation of which as a particle is controversial due to a huge
width of 600--1000 MeV, $f_0(980),$ $f_0(1370)$ (which stands for two separate
states, $f_0(1300)$ and $f_0(1370)$, of a previous edition of PDG \cite{pdg1}),
and $f_0(1500)$ (which also stands for two separate states, $f_0(1525)$ and
$f_0(1590),$ of a previous edition of PDG), and two more possibly scalar
states, the $f_J(1710),$ $J=0$ or 2, seen in radiative $J/\Psi $ decays, and
an $\eta $-$\eta $ resonance $X(1740)$ with uncertain spin, produced in
$p\bar{p}$ annihilation in flight and in charge-exchange. Recently several
groups claimed different scalar isoscalar structures close to 1500 MeV,
including: 1) a narrow state with mass $1445\pm 5$ MeV and width $65\pm 10$
MeV seen by the WA91 collaboration at CERN in central production of $4\pi $ in
high-energy $pp$ collisions \cite{Abatzis}, 2) the lightest of the three
states with masses 1505 MeV, 1750 MeV and 2104 MeV revealed upon reanalyzing
of data on $J/\Psi \rightarrow \gamma 2\pi ^{+}2\pi ^{-}$ \cite{Bugg}, and 3)
the $f_0(1400),$ $f_0(1500),$ $f_0(1520).$ The masses, widths and decay
branching ratios of these states are incompatible within the errors quoted by
the groups. We do not consider it as plausible that so many scalar isoscalar
states exist in such a narrow mass interval. Instead, we take the various
states as manifestation of one object which we identify tentatively with the
$f_0(1450).$
It has been convincingly argued that the narrow $a_0(980)$, which has also
been seen as a narrow structure in $\eta \pi $ scattering, can be generated by
meson-meson dynamics alone \cite{WI,KK}. This interpretation of the $a_0(980)$
leaves the $a_0(1320)$ or $a_0(1450)$ (which may be manifestations of one
state having a mass in the interval 1350-1400 MeV) as the 1 $^3P_0$ $q\bar{q}$
state. Similarly, it is usually assumed that the $f_0(980)$ is a $K\bar{K}$
molecule, as suggested originally by Weinstein and Isgur \cite{WI}. The mass
degeneracy with the $a_0(980)$ and their proximity to the $K\bar{K}$ threshold
seem to require that the nature of both states should be the same. On the
other hand, the $K\bar{K}$ interaction in the $I=1$ and $I=0$ channels is very
different: the extremely attractive $I=0$ interaction may not support a
loosely bound state. Instead, it may just define the pole position of the
$f_0(980)$ $q\bar{q}$ resonance. Indeed, Morgan and Pennington \cite{MP} find
the $f_0(980)$ pole structure characteristic for a genuine resonance of the
constituents and not of a weakly bound system. The $I=1$ $K\bar{K}$ interaction
is weak and may generate a $K\bar{K}$ molecule. Alternatively, T\"{o}rnqvist
\cite{Torn} interprets both the $f_0(980)$ and $a_0(980)$ as the members of the
$q\bar{q}$ nonet with strong coupling to the decay channels. However, this
does not account for the recently discovered $a_0(1320)$ and $a_0(1450).$
With respect to the $f_0(1370)$ (or two separate states, $f_0(1300)$ and $f_0(
1370),$ according to a previous edition of PDG), one may follow the arguments
of Morgan and Pennington \cite{MP} and assume that the $\pi \pi $ interaction
produces both very broad, $f_0(1000),$ and narrow, $f_0(980),$ states, giving
rise to a dip at 980 MeV in the squared $\pi \pi $ scattering amplitude $T_{11
}.$ In this picture, the $f_0(1370)$ is interpreted as the high-mass part of
the $f_0(1000)$ (the low-mass part may be associated with the $\sigma $ of the
most recent PDG). In experiments, the $f_0(1000)$ shows up at $\sim 1300$ MeV
because of the pronounced dip in $|T_{11}|^2$ at $\sim 1$ GeV. The $f_0(1000)$
has an extremely large width; thus a resonance interpretation is
questionable. It could be generated by $t$-channel exchanges instead of
inter-quark forces \cite{ZB}.
The $f_0(1500)$ resonance has been recently observed by the Crystal Barrel
collaboration in $p\bar{p}$ annihilations \cite{CrBa}. It was claimed that
this state has a peculiar decay pattern\footnote{New preliminary results by
Crystal Barrel are \cite{Koch} $$\pi \pi :\eta \eta :\eta \eta ^{'}:K\bar{K}=
1.45:0.34:0.48:0.48\pm 0.24,$$ with normalization of $\pi \pi $ in agreement
with (1).} \cite{AC}
\beq
\pi \pi :\eta \eta :\eta \eta ^{'}:K\bar{K}=1.45:0.39\pm 0.15:0.28\pm 0.12:
<0.15.
\eeq
This pattern can be reproduced by assuming the existence of an additional
scalar state which is mainly $s\bar{s}$ and should have a mass of about 1700
MeV, possibly the $f_J(1710),$ and tuning the mixing of the $f_0(1500)$ with
the $f_0(1370)$ $n\bar{n}$ $(n=u,d)$ and the (predicted) $f_0(1700)$ $s\bar{s}$
states \cite{AC}. In this picture, the $f_0(1500)$ is interpreted as a glueball
state with strong mixing with the close-by conventional scalar mesons.
On the other hand, Lee and Weingarten \cite{LW} interpret the $f_0(1500)$ as a
mainly $s\bar{s}$ state which mixes strongly with the close-by mainly $n\bar{
n}$ and scalar glueball states which show up as resonances at 1390 MeV and 1710
MeV, respectively. The latter is in agreement with the values for the scalar
glueball mass $1740\pm 71$ MeV and $1710\pm 50$ MeV obtained from QCD lattice
calculations by Sexton {\it et al.} \cite{SVW} and Luo {\it et al.} \cite{Luo},
respectively. An interpretation of the $f_0(1500)$ as a conventional $q\bar{q}$
state, as well as a qualitative explanation of its reduced $K\bar{K}$ partial
width, were also given by Klempt {\it et al.} \cite{Klempt} in a relativistic
quark model with linear confinement and instanton-induced interaction. A
quantitative explanation of the reduced $K\bar{K}$ partial width of the $f_0(
1500)$ was given in a recent publication by the same authors \cite{Klempt1}.
The decay pattern obtained in \cite{Klempt1} (with the $f_0$-$f_0^{'}$ mixing
angle $\approx 25^o),$ $$\pi \pi :\eta \eta :\eta \eta ^{'}:K\bar{K}=1.45:0.32
:0.18:0.03,$$ is in excellent agreement with (1).
The above arguments lead one to the following spectrum of the scalar meson
nonet (in the order: isovector, isodoublet, isoscalar mostly octet,
isoscalar mostly singlet),
\beq
a_0(1320)\;\;{\rm or}\;\;a_0(1450),\;\;\;K_0^\ast (1430),\;\;\;f_0(1500),\;\;\;
f_0(980)\;\;{\rm or}\;\;f_0(1000).
\eeq
This spectrum agrees essentially with the $q\bar{q}$ assignments found by
Klempt {\it et al.} \cite{Klempt}, and Dmitrasinovic \cite{Dmitra} who
considered the Nambu--Jona-Lasinio model with a $U_A(1)$ breaking
instanton-induced 't Hooft interaction. The spectrum of the
meson nonet given in \cite{Klempt} is
\beq
a_0(1320),\;K_0^\ast (1430),\;f_0(1470),\;f_0(980),
\eeq
while that suggested by Dmitrasinovic, on the basis of the sum rule
\beq
m_{f_0}^2+m_{f_0^{'}}^2+m_\eta ^2+m_{\eta ^{'}}^2=2(m_K^2+m_{K_0^\ast }^2)
\eeq
derived in his paper, is \cite{Dmitra}
\beq
a_0(1320),\;K_0^\ast (1430),\;f_0(1590),\;f_0(1000).
\eeq
The $q\bar{q}$ assignment obtained by one of the authors by the application of
the linear mass spectrum discussed in ref. \cite{linear} to a composite system
of the two, pseudoscalar and scalar nonets, is \cite{invited}
\beq
a_0(1320),\;K_0^\ast (1430),\;f_0(1525),\;f_0(980),
\eeq
in essential agreement with (3) and (5). The assignment (6) has found further
justification in the constituent quark model explored by us in ref.
\cite{P-wave}.
In our previous paper \cite{BG}, by using Regge phenomenology, we derived a
new mass relation $(I$ stands for isospin):
\beq
2M^2(s\bar{s})+3M^2(n\bar{n},I=1)=4M^2(s\bar{n})+M^2(n\bar{n},I=0).
\eeq
This relation was obtained for the pseudoscalar mesons, and further generalized
to every meson multiplet. It differs from the Sakurai mass formula
\cite{Sakurai} obtained in the case of the ideal nonet mixing,
\beq
\theta _{id}=\arctan \frac{1}{\sqrt{2}}\approx 35.3^o,
\eeq
\beq
2M^2(s\bar{s})+M^2(n\bar{n},I=1)+M^2(n\bar{n},I=0)=4M^2(s\bar{n})
\eeq
by only a term which depends explicitly on isospin variation, $\sim (M^2(n\bar{
n},I=1)-M^2(n\bar{n},I=0)).$ This term improves the accuracy of the Sakurai
formula, which is not bad by itself, $(\sim $2-3\%) by a factor of 2 \cite{BG}.
In this paper we present another derivation of the formula (7) for
pseudoscalar and scalar mesons in the Nambu--Jona-Lasinio model with an
instanton-induced interaction. Before doing this, let us mention that this
formula can be applied to the physical states of these two nonets provided the
mixing angles are known. For example, for pseudoscalar mesons, the
$\eta $-$\eta ^{'}$ mixing angle is given by duality constraints \cite{Bra},
\beq
\tan \theta _{\eta \eta ^{'}}=-\frac{1}{2\sqrt{2}},\;\;\;\theta _{\eta \eta ^{
'}}\approx -19.5^o,
\eeq
in good agreement with most of experimental data \cite{data}. In view of the
relations
$$\left(
\begin{array}{c}
\eta \\
\eta ^{'}
\end{array}
\right) =\left(
\begin{array}{lr}
\cos \theta _{\eta \eta ^{'}} & -\sin \theta _{\eta \eta ^{'}} \\
\sin \theta _{\eta \eta ^{'}} & \cos \theta _{\eta \eta ^{'}}
\end{array}
\right) \left(
\begin{array}{c}
\eta _8 \\
\eta _9
\end{array}
\right) ,\;\;\left(
\begin{array}{c}
\eta _s \\
\eta _n
\end{array}
\right) =\left(
\begin{array}{lr}
\cos \theta _{id} & -\sin \theta _{id} \\
\sin \theta _{id} & \cos \theta _{id}
\end{array}
\right) \left(
\begin{array}{c}
\eta _8 \\
\eta _9
\end{array}
\right) ,$$ where
$$\eta _8=\frac{u\bar{u}+d\bar{d}-2s\bar{s}}{\sqrt{6}},\;\;\;\eta _9=\frac{
u\bar{u}+d\bar{d}+s\bar{s}}{\sqrt{3}}$$ are the isoscalar octet and singlet
states, respectively, and
$$\eta _n=\frac{u\bar{u}+d\bar{d}}{\sqrt{2}},\;\;\;\eta _s=s\bar{s}$$ are the
``ideal-mixture'' counterparts of the physical $\eta $ and $\eta ^{'}$ states,
one obtains
$$\left(
\begin{array}{c}
\eta _s \\
\eta _n
\end{array}
\right) =\left(
\begin{array}{lr}
\cos (\theta _{id}-\theta _{\eta \eta ^{'}}) & -\sin (\theta _{id}-\theta _{
\eta \eta ^{'}}) \\
\sin (\theta _{id}-\theta _{\eta \eta ^{'}}) & \cos (\theta _{id}-\theta _{
\eta \eta ^{'}})
\end{array}
\right) \left(
\begin{array}{c}
\eta \\
\eta ^{'}
\end{array}
\right) $$
\beq
=\left(
\begin{array}{lr}
\cos \xi & -\sin \xi \\
\sin \xi & \cos \xi
\end{array}
\right) \left(
\begin{array}{c}
\eta \\
\eta ^{'}
\end{array}
\right) ,
\eeq
where $\xi \equiv \theta _{id}-\theta _{\eta \eta ^{'}}$ and, as follows from
(8),
\beq
\cos \xi =\frac{\sin \theta _{\eta \eta ^{'}}+\sqrt{2}\cos \theta _{\eta \eta
^{'}}}{\sqrt{3}}.
\eeq
Assuming, as usual, that the relevant matrix elements are equal to the squared
masses of the corresponding states, and using the orthogonality of the $\eta $
and $\eta ^{'}$ as physical states, we obtain, using (10)-(12),
\bqry
m^2_{\eta _n} & = & \frac{2}{3}\;m^2_{\eta }\;+\;\frac{1}{3}\;m^2_{\eta ^{'}},
\\
m^2_{\eta _s} & = & \frac{1}{3}\;m^2_{\eta }\;+\;\frac{2}{3}\;m^2_{\eta ^{'}},
\eqry
in agreement with naive expectations from the quark content of these states
which is, in view of (10) \cite{BG},
$$\eta =\frac{u\bar{u}+d\bar{d}-s\bar{s}}{\sqrt{3}},\;\;\;\eta ^{'}=\frac{
u\bar{u}+d\bar{d}+2s\bar{s}}{\sqrt{6}}.$$
The use of the values (13),(14) in Eq. (7) leads finally to
\beq
4m_K^2=3m_\pi ^2+m_{\eta ^{'}}^2,
\eeq
which is the new Gell-Mann--Okubo mass formula for pseudoscalar mesons found in
our previous publication \cite{BG} and which is satisfied to an accuracy of
better than 1\% by the measured pseudoscalar meson masses.
We now turn to the derivation of the formula (7). We shall adopt the version
of the Nambu--Jona-Lasinio (NJL) model which includes the U$_A(1)$ breaking 't
Hooft interaction \cite{tH} in the form of a $2N_f$-point determinant nonlocal
quark interaction, where $N_f$ is the number of flavors. This model has been
extensively studied by Dmitrasinovic \cite{Dmitra,Dmitra2}. For $N_f=2,$
and in the local interaction limit, the sum of the determinant and its
Hermitian conjugate is equivalent to the following four-point interaction:
\bqry
L_{tH}^{(4)} & = & G_2\Big[ {\rm det}\left( \bar{\psi }(1+\gamma _5)\psi
\right) +{\rm H.c.}\Big] \NL
& = & \frac{G_2}{2}\Big[ \left( \bar{\psi }\psi \right) ^2-\left( \bar{\psi }
\mbox{\boldmath $\tau $}\psi \right) ^2-\left( \bar{\psi }i\gamma _5\psi
\right) ^2+\left( \bar{\psi }i\gamma _5\mbox{\boldmath $ \tau $}\psi \right)
^2\Big] .
\eqry
Therefore, the resulting effective Lagrangian is
\bqry
L_{NJL}^{(4)} & = & L_{NJL}\;+\;L_{tH}^{(4)}\;=\;\bar{\psi }\left[ i\gamma
\partial -m^0\right] \psi \NL
& & +\;\frac{G_1}{2}\Big[ \left( \bar{\psi }\psi \right) ^2+\left( \bar{
\psi }\mbox{\boldmath $\tau $}\psi \right) ^2+\left( \bar{\psi }i\gamma _5\psi
\right) ^2+\left( \bar{\psi }i\gamma _5\mbox{\boldmath $\tau $}\psi \right) ^2
\Big] \NL
& & +\;\frac{G_2}{2}\Big[ \left( \bar{\psi }\psi \right) ^2-\left( \bar{
\psi }\mbox{\boldmath $\tau $}\psi \right) ^2-\left( \bar{\psi }i\gamma _5\psi
\right) ^2+\left( \bar{\psi }i\gamma _5\mbox{\boldmath $\tau $}\psi \right) ^2
\Big] .
\eqry
The coupling constants $G_1$ and $G_2$ scale as $$G_1=O\left( \frac{1}{N_c}
\right) ,\;\;\;G_2=O\left( \frac{1}{N_c^2}\right) $$ in the large $N_c$ limit,
$N_c$ being the number of colors \cite{Dmitra}. Note that in the point limit
of single gluon exchange, QCD would produce the $G_1$ temrs only (as well as
their vector and axial-vector analogs) \cite{GH}.
The original NJL model contains the U$_L(2)\times $U$_R(2)$-symmetric and the
U$_A(1)$ breaking terms with equal weights in the Lagrangian, i.e., with $G_1=
G_2.$ This case corresponds to the complete vanishing of interaction in the
isoscalar pseudoscalar and isovector scalar channels, and hence to the complete
disappearance of these states from the spectrum of the model, as seen in Eq.
(17). This situation is referred to as ``maximal U$_A(1)$ breaking'' in ref.
\cite{Dmitra}. Since isoscalar pseudoscalar mesons $\eta $ and $\eta ^{'}$ do
exist and one can construct the ideal mixture of them, i.e., a linear
combination of the two that contains no $s\bar{s}$ component (and corresponds
to the isoscalar pseudoscalar state of the two-flavor version of NJL), one has
to relax the severity of U$_A(1)$ breaking in the model and thus consider the
``minimally extended'' \cite{Dmitra} two-flavor NJL model, Eq. (17), with $G_1
\neq G_2.$
Working out this model in a standard manner, one finds the familiar NJL gap
equation for the constituent quark mass $m$ \cite{Klev},
\bqry
m & = & m^0\;-\;\left( G_1+G_2\right) \langle \bar{\psi }(x)\psi (x)\rangle _0
\NL
& = & m^0\;+\;4iN_cN_f\left( G_1+G_2\right) \int \frac{d^4p}{(2\pi )^4}\;
\frac{m}{p^2-m^2},
\eqry
which has to be regularized either by introducing a Euclidean cutoff or
following Pauli and Villars (PV) \cite{PV}. The self-consistency condition
$\Sigma _H=m,$ where $\Sigma _H$ is the quark self-energy, determines $m$ at
the one-loop level. This $m$ is also related to the quark condensate:
$$ m=m^0-\left( G_1+G_2\right) \langle \bar{\psi }(x)\psi (x)\rangle _0.$$
A nonzero value of $m$ in the chiral limit $m^0=0$ signals the breakdown of
chiral symmetry.
The meson masses are further read off from the poles of the corresponding
propagators of the inhomogeneous Bethe-Salpeter equation describing the
scattering of quarks and antiquarks which are
\beq
-iD(k)=\frac{i(G_1+G_2)}{1-(G_1+G_2)\Pi (k)},
\eeq
where $k$ is the four-momentum transfer, and $\Pi (k)$ represents the sum of
all proper polarization diagrams in the relevant channel. The form of the
interaction in Eq. (17) gives rise to scattering in the four channels: the
isovector pseudoscalar $(\pi ),$ isoscalar pseudoscalar $(\eta _n),$
isoscalar scalar $(\sigma )$ and isovector scalar $(\mbox{\boldmath $\sigma
$}).$ One finds, e.g. \cite{Klev},
\beq
\Pi _\pi (k)-\frac{\Sigma _H}{(G_1+G_2)m}=-4iN_ck^2I(k),
\eeq
where $I(k)$ is a logarithmically divergent loop integral,
\beq
I(k)=\int \frac{d^4p}{(2\pi )^4}\;\frac{1}{[p^2-m^2][(p+k)^2-m^2]},
\eeq
which is PV-regularized. It follows from (20) that
$$\Sigma _H=\left( G_1+G_2\right) m\Pi _\pi (0)$$ which, when combined with
$\Sigma _H=m,$ leads to the self-consistency condition
\beq
1=\left( G_1+G_2\right) \Pi _\pi (0).
\eeq
Then $D$ of Eq. (19) for the pion mode becomes, through (20) \cite{Dmitra}
\bqry
-iD_\pi (k) & = & \frac{i(G_1+G_2)}{1-(G_1+G_2)\Pi _\pi (k)}=\frac{1}{4N_cI(
k)(k^2+i\varepsilon )} \NL
& = & \frac{-ig^2_{\pi qq}}{(k^2+i\varepsilon )F(k)},
\eqry
and similarly, for the remaining three modes \cite{Dmitra},
\bqry
-iD_\sigma (k) & = & \frac{i(G_1+G_2)}{1-(G_1+G_2)\Pi _\sigma (k)}\;=\;\frac{
-ig^2_{\pi qq}}{(k^2-4m^2+i\varepsilon )F(k)}, \\
-iD_{\eta _n}(k) & = & \frac{i(G_1-G_2)}{1-(G_1-G_2)\Pi _\pi (k)}\;=\;\frac{
-ig^2_{\pi qq}}{(k^2+i\varepsilon )F(k)-m^2_{tH}}, \\
-iD_{\mbox{\boldmath $\sigma $}}(k) & = & \frac{i(G_1-G_2)}{1-(G_1-G_2)\Pi _
\sigma (k)}\;=\;\frac{-ig^2_\pi qq}{(k^2-4m^2+i\varepsilon )F(k)-m^2_{tH}},
\eqry
where
\beq
m^2_{tH}\left( N_f=2\right) \;\equiv \;\frac{2g^2_{\pi qq}G_2}{G_1^2-G_2^2}\;
\simeq \;\frac{2g^2_{\pi qq}G_2}{G_1^2}\;\!+\;\!O\left( \frac{1}{N_c^2}\right)
\eeq
is the 't Hooft mass \cite{Dmitra}, and the associated zero external momentum
coupling constants are \cite{Dmitra}
\bqry
g^2_{\pi qq} & = & g^2_{\sigma qq}\;=\;g^2_{\eta _nqq}\;=\;g^2_{\mbox{
\boldmath $\sigma $}qq}\;=\;\left( \frac{\partial \Pi }{\partial k^2}\right) ^{
-1}_0 \NL
& = & \Big[ -4iN_cI(0)\Big] ^{-1}\;=\;\left( \frac{m}{f_\pi }\right) ^2,
\eqry
where $f_\pi =93$ MeV is the pion decay constant. In Eqs. (23)-(27), $F(k)=I(
k)/I(0).$ This factor $F(k)$ provides the $D$'s with a more complicated
analytic structure than that of the free scalar propagator; this reflects the
composite nature of the bound states which they describe. However, as
discussed in detail in ref. \cite{Dmitra}, one may set $F(k)=1$ as a reliable
approximation which does not affect the predictions of the model essentially.
With $F(k)=1,$ reading off the poles of the corresponding $D$'s in (23)-(27),
and switching to the hadron spectroscopy notations $$f_{0n}=\sigma ,\;\;\;a_0=
\mbox{\boldmath $\sigma $},$$ one finds the following meson masses, upon
introducing explicit chiral symmetry breaking in the form of nonzero current
quark masses $m^0$ \cite{Dmitra}:
\bqry
m_{\eta _n}^2 & = & m_\pi ^2\;+\;m_{tH}^2\left( N_f=2\right) , \\
m_{a_0}^2 & = & m_\pi ^2\;+\;4m^2\;+\;m_{tH}^2\left( N_f=2\right) , \\
m_{f_{0n}}^2 & = & m_\pi ^2\;+\;4m^2,
\eqry
which lead to the mass relation
\beq
m_{a_0}^2-m_{f_{0n}}^2=m_{\eta _n}^2-m_\pi ^2=m_{tH}^2\left( N_f=2\right) ,
\eeq
found first by Dmitrasinovic \cite{Dmitra}, and which is a direct consequence
of U$_A(1)$ breaking in this NJL model (by instanton-induced 't Hooft
interaction). Here $\eta _n$ and $f_{0n}$ are the nonstrange ideal mixtures of
the $\eta ,\eta ^{'}$ and $f_0,f_0^{'}$ mesons, respectively, in the badly
broken SU(3) limit.
It can now be easily understood how ``maximal breaking'' of U$_A(1)$ occurs in
this NJL model. It is reached in the limit of equal coupling constants, $G_1=
G_2.$ In this limit, in which the simplest NJL Lagrangian is recovered, as
seen in Eq. (17), the 't~Hooft mass, Eq. (27), and the masses of $\eta _n$ and
$a_0,$ go to infinity. Hence, these heavy modes cannot propagate, so that they
completely decouple from the model. That explains their absence in the original
NJL model.
The three-flavor generalization of this NJL model is straightforward: The free
Lagrangian and the U$(3)_L\times $U$(3)_R$-symmetric quartic self-interaction
terms are essentially the same as in Eq. (17), the number of terms being
appropriately extended to 18:
\bqry
L_{NJL}^{(6)} & = & \bar{\psi }\left[ i\gamma \partial -m^0\right] \psi \;+\;
G\sum _{i=0}^8\Big[ \left( \bar{\psi }\mbox{\boldmath $\lambda $}_i\psi \right)
^2+\left( \bar{\psi }i\gamma _5\mbox{\boldmath $\lambda $}_i\psi \right) ^2
\Big] \NL
& & -\;K\Big[ {\rm det}\left( \bar{\psi }(1+\gamma _5)\psi \right) +{\rm
det}\left( \bar{\psi }(1-\gamma _5)\psi \right) \Big] ,
\eqry
where $\mbox{\boldmath $\lambda $}_i$ are the SU(3) Gell-Mann structure
constants, but the U$_A(1)$ breaking determinant interaction term is now of
sixth order in the Fermi fields, rather than of fourth order as in the $N_f=2$
case. Since one cannot work directly with a sixth-order operator, one has to
construct an effective mean-field quartic Lagrangian using the expectation
values of $\bar{\psi }\psi $ \cite{Klev,BJM}. In the $SU(3)$-symmetric limit,
$\langle \bar{\psi }\mbox{\boldmath $\lambda $}_0\psi \rangle =\sqrt{2/3}\;\!
\langle \bar{\psi }\psi \rangle \neq 0,$ $\langle \bar{\psi }\mbox{\boldmath
$\lambda $}_3\psi \rangle =\langle \bar{\psi }\mbox{\boldmath $\lambda $}_8
\psi \rangle =0,$ one finds the following effective Lagrangian \cite{Klev}:
\bqry
L_{eff}^{(4)} & = & \bar{\psi }\left[ i\gamma \partial -m^0\right] \psi \;+\;
\Big[ K_0^{(-)}\left( \bar{\psi }\mbox{\boldmath $\lambda $}_0\psi \right) ^2+
\sum _{i=1}^8K_i^{(+)}\left( \bar{\psi }i\gamma _5\mbox{\boldmath $\lambda $}_i
\psi \right) ^2\Big] \NL
& & +\;\Big[ K_0^{(+)}\left( \bar{\psi }i\gamma _5\mbox{\boldmath $\lambda
$}_0\psi \right) ^2+\sum _{i=1}^8K_i^{(-)}\left( \bar{\psi }\mbox{\boldmath
$\lambda $}_i\psi \right) ^2\Big] ,
\eqry
where
\bqry
K_0^{(\pm )} & = & G\;\pm \;K\langle \bar{q}q\rangle \;=\;\frac{1}{2}\left( G_
1\;\mp \;2G_2\right) , \\
K_i^{(\pm )} & = & G\;\mp \;\frac{1}{2}K\langle \bar{q}q\rangle \;=\;\frac{1}{
2}\left( G_1\;\pm \;G_2\right) ,\;\;\;i=1,2,\ldots ,8,
\eqry
i.e.,
\beq
G_1=2G,\;\;\;G_2=-\frac{1}{3}K\langle \bar{q}q\rangle ,
\eeq
where the quark condensates are defined as
\bqry
\langle \bar{q}q\rangle & = & -iN_c\;{\rm Tr}\;\!S_F(x,x)^q\;=\;-4iN_c\int
\frac{d^4p}{(2\pi )^4}\;\frac{m_q}{p^2-m_q^2+i\varepsilon },\;\;\;q=u,d,s, \NL
\langle \bar{\psi }\psi \rangle & = & \langle \bar{u}u\rangle +\langle \bar{d}
d\rangle +\langle \bar{s}s\rangle \;=\;-iN_c\;{\rm Tr}\;\!S_F(x,x).
\eqry
As in the $N_f=2$ case, the meson masses are read off from the poles of the
corresponding propagators which are in turn constrained by the system of the
gap equations \cite{Dmitra}:
\bqry
m_u & = & m_u^0\;-\;4G\langle \bar{u}u\rangle \;+\;2K\langle \bar{d}d\rangle
\langle \bar{s}s\rangle , \NL
m_d & = & m_d^0\;-\;4G\langle \bar{d}d\rangle \;+\;2K\langle \bar{s}s\rangle
\langle \bar{u}u\rangle , \NL
m_s & = & m_s^0\;-\;4G\langle \bar{s}s\rangle \;+\;2K\langle \bar{u}u\rangle
\langle \bar{d}d\rangle .
\eqry
In the chiral limit when the bare (current) masses $m_q,\;q=u,d,s$ vanish, and
in the SU(3)-symmetric limit where $m_u=m_d=m_s,$ one has $\langle \bar{u}u
\rangle =\langle \bar{d}d\rangle =\langle \bar{s}s\rangle =\langle \bar{q}q
\rangle ,$ and hence the system (39) decouples:
\beq
m_q=m_q^0-4G\langle \bar{q}q\rangle +2K\langle \bar{q}q\rangle ^2=m_q^0-4K_1^{
(+)}\langle \bar{q}q\rangle .
\eeq
Upon introducing explicit chiral breaking in the form of nonzero current quark
masses $m_i^0,$ we find the following results from Eq. (40) and the effective
Lagrangian (34) \cite{Dmitra}:
\bqry
m_{a_0}^2 & = & m_\pi ^2\;+\;4m^2\;+\;\frac{2}{3}\;\!m_{tH}^2\left( N_f=3
\right) , \\
m_{K^\ast _0}^2 & = & m_K^2\;+\;4m^2\;+\;\frac{2}{3}\;\!m_{tH}^2\left( N_f=3
\right) , \\
m_{\eta }^2\;+\;m_{\eta ^{'}}^2 & = & 2m_K^2\;+\;m_{tH}^2\left( N_f=3\right) ,
\\
m_{f_0}^2\;+\;m_{f_0^{'}}^2 & = & 2m_{K_0^\ast }^2\;-\;m_{tH}^2\left( N_f=3
\right) , \\
m_{f_0}^2\;+\;m_{f_0^{'}}^2 & = & m_{\eta }^2\;+\;m_{\eta ^{'}}^2\;+\;8m^2\;-\;
\frac{2}{3}\;\!m_{tH}^2\left( N_f=3\right) ,
\eqry
where
\beq
m^2_{tH}\left( N_f=3\right) \;\equiv \;\frac{3g^2_{\pi qq}G_2}{G_1^2-G_2^2}\;=
\;\frac{3}{2}\;\!m^2_{tH}\left( N_f=2\right) \;\simeq \;\frac{3g^2_{\pi qq}G_
2}{G_1^2}\;\!+\;\!O\left( \frac{1}{N_c^2}\right) ,
\eeq
in view of (27).
Summing up Eqs. (44) and (45), one immediately obtains the Dmitrasinovic sum
rule (4). This sume rule is not, however, the only consequence of this model.
Indeed, Eqs. (32),(43),(46) and $m_{\eta }^2+m_{\eta ^{'}}^2=m_{\eta _n}^2+m_{
\eta _s}^2$ which follows in general from (11), and in particular, with the
mixing angle (10), from (13),(14), lead to
\bqry
m_{\eta _n}^2\;-\;m_\pi ^2 & = & \frac{2}{3}\;\!m_{tH}^2\left( N_f=3\right) ,
\\
m_{\eta _n}^2\;+\;m_{\eta _s}^2 & = & 2m_K^2\;+\;m^2_{tH}\left( N_f=3\right) ,
\eqry
which, upon eliminating $m_{tH}^2(N_f=3),$ yields
\beq
2m_{\eta _s}^2+3m_\pi ^2=4m_K^2+m_{\eta _n}^2.
\eeq
Similarly, Eqs. (32),(44),(46) and
\beq
m_{f_0}^2+m_{f_0^{'}}^2=m_{f_{0n}}^2+m_{f_{0s}}^2,
\eeq
which follows from Eqs. (55),(56) below, lead to
\bqryn
m_{a_0}^2\;-\;m_{f_{0n}}^2 & = & \frac{2}{3}\;\!m_{tH}^2\left( N_f=3\right) ,
\\
m^2_{f_{0n}}\;+\;m^2_{f_{0s}} & = & 2m_{K_0^\ast }^2\;-\;m^2_{tH}\left( N_f=3
\right) ,
\eqryn
which again, upon eliminating $m_{tH}^2(N_f=3),$ yields
\beq
2m^2_{f_{0s}}+3m_{a_0}^2=4m_{K_0^\ast }^2+m^2_{f_{0n}}.
\eeq
Eqs. (49) and (51) represent the formula (7) written for the pseudoscalar and
scalar mesons, respectively.
We consider the fact that the formula (7) can be derived in (at least) two
completely independent ways, viz., from Regge phenomenology, as done in ref.
\cite{BG}, and in the NJL model, as done in the present paper, as sufficient
to treat this formula as (almost) model-independent.\footnote{We know of at
least four other distinct approaches which also yield these same results.}
As discussed above, in the pseudoscalar meson case, this formula reduces to
Eq. (15) which is satisfied to an accuracy of better than 1\%; it therefore
finds its direct experimental confirmation in terms of the pseudoscalar meson
mass spectrum. Since this formula should describe the physical scalar meson
mass spectrum as well, we shall apply it to the problem of the correct $q\bar{
q}$ assignment for the scalar meson nonet.
We note first that the masses of the ideally mixed counterparts of the physical
$f_0$ and $f_0^{'}$ mesons are easily obtained from Eqs.
(44),(49),(51):\footnote{Similar relations for the pseudoscalar mesons follow
from Eqs. (47),(48) above.}
\bqry
m^2_{f_{0n}} & = & m_{a_0}^2\;-\;\frac{2}{3}\;\!m_{tH}^2\left( N_f=3\right) ,
\\
m^2_{f_{0s}} & = & 2m_{K_0^\ast }^2\;-\;m_{a_0}^2\;-\;\frac{1}{3}\;\!m_{tH}^2
\left( N_f=3\right) ,
\eqry
where the numerical value of $m^2_{tH}(N_f=3)$ is
\beq
m^2_{tH}\left( N_f=3\right) =m_{\eta }^2+m_{\eta ^{'}}^2-2m_K^2\cong 0.72\;{\rm
GeV}^2.
\eeq
To determine the masses of the physical $f_0$ and $f_0^{'}$ states, one has to
know the $f_0$-$f_0^{'}$ mixing angle.
In the pseudoscalar meson case, the $\eta $-$\eta ^{'}$ mixing angle is given
by the mass-matrix diagonalization as \cite{BG}
\beq
\tan 2\theta _{\eta \eta ^{'}}=2\sqrt{2}\left( 1-\frac{3m^2_{tH}(N_f=3)}{2(m_
K^2-m_\pi ^2)}\right) ^{-1}\approx -19^o,
\eeq
in agreement with (10). In the scalar meson case, one has,
respectively,\footnote{The formula (56) coincides with Eq. (51) of ref.
\cite{Dmitra}, and Eq. (41) of ref. \cite{Dmitra2}.}
\beq
\tan 2\theta _{f_0f_0^{'}}=2\sqrt{2}\left( 1+\frac{3m^2_{tH}(N_f=3)}{2(m_
{K_0^\ast }^2-m_{a_0}^2)}\right) ^{-1},
\eeq
since the sign of the instanton-induced 't Hooft interaction (the sign of $m^2
_{tH})$ for the scalar mesons is opposite to that for the pseudoscalar mesons,
as is clear from the above consideration of the NJL model.
As seen in Eqs. (47),(48),(52),(53),(55),(56), when the 't Hooft interaction
is switched off, $m^2_{tH}(N_f=3)=0,$ one recovers ideal mixing, $\theta _{
\eta \eta ^{'}}=\theta _{f_0f_0^{'}}=\arctan 1/\sqrt{2},$ and the
corresponding ideal structures for both multiplets,
$$m^2_{\eta }=m^2_{\eta _s}=2m_K^2-m_\pi ^2,\;\;\;m^2_{\eta ^{'}}=m^2_{\eta _
n}=m^2_\pi ,$$ $$m^2_{f_0}=m^2_{f_{0s}}=2m_{K_0^\ast }^2-m_{a_0}^2,\;\;\;m^2_{
f_0^{'}}=m^2_{f_{0n}}=m^2_{a_0},$$
which are solutions to Eqs. (47) and (49), respectively, as they should be.
Using Eqs. (52),(53),(56) and
\bqry
m^2_{f_{0n}} & = & \sin ^2\xi \;m^2_{f_0}\;+\;\cos ^2\xi \;m^2_{f_0^{'}}, \\
m^2_{f_{0s}} & = & \cos ^2\xi \;m^2_{f_0}\;+\;\sin ^2\xi \;m^2_{f_0^{'}},
\eqry
\beq
\cos \xi =\frac{\sin \theta _{f_0f_0^{'}}+\sqrt{2}\cos \theta _{f_0f_0^{'}}}{
\sqrt{3}},
\eeq
which follow from relations similar to Eqs. (11),(12) in the scalar meson
case, one can easily calculate the masses of the $f_0$ and $f_0^{'}$ states
and their mixing angle.
The results of our calculation are presented in Table I where we have
considered a number of different possibilities for the mass of the isovector
scalar state $a_0$ consistent with the masses of the experimental candidates
$a_0(980),$ $a_0(1320)$ and $a_0(1450).$
\begin{center}
\begin{tabular}{|c|c|c|c|} \hline
$m_{a_0},$ MeV & $m_{f_0},$ MeV & $m_{f_0^{'}},$ MeV & $\theta _{f_0f_0^{'}},$
degrees \\ \hline \hline
983.5 & 1710 & 663 & 27.3 \\ \hline \hline
1290 & 1511 & 1039 & 18.1 \\ \hline
1320 & 1490 & 1069 & 15.8 \\ \hline
1350 & 1471 & 1096 & 12.8 \\ \hline \hline
1410 & 1437 & 1140 & 3.8 \\ \hline
1450 & 1424 & 1156 & -4.8 \\ \hline
1490 & 1423 & 1157 & -14.6 \\ \hline
\end{tabular}
\end{center}
{\bf Table I.} Predictions for the masses and mixing angle of the scalar mesons
for different values of the $a_0$ meson mass, according to Eqs.
(52),(53),(56)-(59). The value of the $K_0^\ast $ meson mass used in the
calculation is $m_{K_0^\ast }=1429$ MeV. \\
For the $a_0(980)$ taken as the isovector scalar state, our solution exhibits
a number of interesting features: It predicts an $f_0$-$f_0^{'}$ mixing angle
close to the ideal mixture value (8), i.e., both isoscalar states would be
almost pure $n\bar{n}$ and $s\bar{s}$; their masses are predicted to be in
agreement with those of the $\sigma $ of the most recent edition of PDG and
the $f_J(1710),$ respectively. The existence of the latter state, which in
this case is almost pure $s\bar{s},$ would be in agreement with the prediction
by Amsler and Close \cite{AC}. However, we will disregard this solution as
unphysical. Indeed, as we discuss at the beginning of the paper, convincing
arguments exist in the literature regarding the $a_0(980)$ not being a $q\bar{
q}$ state, but rather molecular in character. The existence of the
$\sigma $-meson (to be identified as the $f_0^{'}(663)$ in this case) is
controversial due to its enormous width. The spin of the $f_J(1710)$ meson is
still uncertain $(J=0$ or 2). Moreover, with this solution, the mass interval
occupied by the scalar nonet turns out to be wider than 1 GeV, a typical mass
scale of light hadron spectroscopy, which cannot be considered as plausible.
Thus, we restrict ourselves to the solution with $m_{a_0}=1390\pm 100$ MeV (to
accommodate the both experimental candidates $a_0(1320)$ and $a_0(1450))$
which, according to the results presented in Table I, is
$$m_{f_0}=1467\pm 44\;{\rm MeV,}\;\;\;m_{f_0^{'}}=1098\pm 59\;{\rm MeV,}\;\;\;
\theta _{f_0f_0^{'}}=1.8\pm 16.3^o.$$
It is interesting to note that for rather wide ranges of the $a_0$ mass and
$f_0$-$f_0^{'}$ mixing angle, the masses of two isoscalar states are
accurately predicted to be in the vicinity of 1.1 GeV and 1.45 GeV. Especially
if the $a_0(1450)$ is the true isovector scalar state (which we hope will be
either confirmed or refuted by ongoing experiments), then for a wide range of
its mass $(1450\pm 40$ MeV) the solution picks the masses of the two isoscalars
very accurately at $1430\pm 7$ MeV and $1148\pm 8$ MeV. Nothing can be said,
however, about their decay properties so far, since the solution, albeit
specifying the two masses, cannot choose between the mixing angles, the proper
determination of which requires a specific value of the $a_0$ mass.
The solution that we obtained provides the mass of the scalar isoscalar mostly
singlet state around 1.1 GeV. This value may be considered as the pole position
which gets shifted further down due to strong coupling to, e.g., the $K\bar{
K}$ threshold. For example, recent analysis of various experimental data by
Anisovich {\it et al.} \cite{Anis} reveals a lowest mass scalar resonance with
the pole position at $1015\pm 15$ MeV and width of $43\pm 8$ MeV. We do not
address the question of choosing between the $f_0(980)$ and $f_0(1000)$ in this
paper. Let us only mention that with the $f_0$-$f_0^{'}$ mixing angle $\theta
_{f_0f_0^{'}}=-1/2\;\arctan 1/(2\sqrt{2})\approx -9.7^o$ which is achieved with
$m_{a_0}\approx 1470$ MeV, the quark content of the two isoscalars turn out to
be $$f_0=\frac{n\bar{n}-s\bar{s}}{\sqrt{2}},\;\;\;f_0^{'}=\frac{n\bar{n}+s\bar{
s}}{\sqrt{2}},\;\;\;n\bar{n}\equiv \frac{u\bar{u}+d\bar{d}}{\sqrt{2}},$$ i.e.,
the both have (50\% $n\bar{n},$ 50\% $s\bar{s})$ quark content. Such quark
content can, e.g., easily explain a large branching ratio of 22\% of the $f_0(
980)$ decay into the $K\bar{K}$ channel.
The mass of the scalar isoscalar mostly octet state predicted by our solution,
$1467\pm 44$ MeV, is in good agreement with recent analysis by Kaminski {\it
et al.} \cite{KLL} of the three coupled channel $(\pi \pi,\;K\bar{K},\;4\pi )$
model which reveals a relatively narrow $(\Gamma =135\pm 45$ MeV) resonance
with a mass of $1430\pm 30$ MeV.
We remark also that the masses of both isoscalar scalar states predicted in
this paper are below a typical range of $1600\pm 100$ MeV provided by QCD
lattice calculations as that for the lightest scalar glueball
\cite{SVW,Luo,lat}. They may however get shifted from the predicted values
because of possible mixture with the lightest scalar glueball.
The $q\bar{q}$ assignment found in this work,
\beq
a_0(1320)\;\;{\rm or}\;\;a_0(1450),\;\;\;K_0^\ast (1430),\;\;\;f_0(1450),\;\;\;
f_0(980)\;\;{\rm or}\;\;f_0(1000),
\eeq
is consistent with the assignments (3),(5),(6) established in three different
approaches. The fact that different approaches lead to essentially the same
result encourage us to believe that we are not far from the complete resolution
of the scalar meson nonet enigma.
\bigskip
\bigskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 782 |
In keeping with tradition (thank you Gail!), I'd like to offer you free priority shipping today on all orders from my shop over $36 (I may have to rethink this whole idea once I start having major issues with my age...I feel it coming). This is a great time to stock up on pie boxes for gifting, Thanksgiving leftovers, bake sales, an upcoming wedding, etc. In fact, if pie isn't your thing, fill these sweet boxes with a slice of cake...or quiche...or a tart you picked up from your favorite bakery. Surely you will make someone's day, which is one of the best feelings in the world.
I'm planning on running this special through midnight PST, so you've got almost exactly 24 hours (from the time I publish this post), to stock up on original packaging you can only find at Petit Moulin on Etsy. As always, thank you for your support and have a wonderful day today wherever you may be!
images by me, Stephanie Brubaker for stephmodo - thanks for stopping by!
Surely you will make someone's day, which is one of the best feelings in the world.
If I'm reading things right, Happy Birthday!
A lovely tradition. Thank you! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,234 |
From time to time the ALD will publish position papers and statements of interest to the dental professional to assist in your understanding of the role of dental lasers in healthcare. These statements and papers are typically developed by ALD's Science and Research Committee, approved by the Board of Directors and published in the Journal of Laser Dentistry. Each paper or statement may be revised and updated as technology changes and improves. The viewer is encouraged to visit often for updates and additional postings.
Note: Papers and Articles are copyright protected by the authors and the ALD and may not be reproduced in any form. Requests for reprint permission may be sent to the Office of the Executive Director by email laserexec@laserdentistry.org. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,666 |
{"url":"http:\/\/www.eskesthai.com\/2007\/01\/latex-rendering.html","text":"## Wednesday, January 03, 2007\n\n### Latex Rendering\n\nAndrew Roberts:Here are some tutorials I have written for getting up to speed with this excellent document processing system. If you are not sure why Latex is any good, find out the benefits. I wouldn't consider myself an expert, but I'm learning all the time. I recall finding it quite taxing when I start to learn Latex, which is why I have started these tutorials. However, I hope that my experiences plays to your advantage, since I hope I can let you into the sort of questions and problems I had when I first learning Latex.\n\nWhile preparing myself for the intricacies of PDF or science documents, which is a requirement of science people, I went back of course to what Robert and Clifford had found in the science community.\n\nI am of course least in terms of the science education as I go through these bloggeries, yet the message has not fallen on a deaf ear in my case. So working toward helping others in science, is no less important then being given the skill to express myself\/yourself properly.\n\nSo this is a beginning for me then, of what I had said I would do for the coming year.\n\nLatex Symbol Latex language{times} \\times{div} \\div{diamond} \\diamond{pm} \\mp{ominus} \\ominus \\Big oplus{otimes} \\otimes{oslash} \\oslash{odot} \\odot{bigcirc} \\bigcirc{circ} \\circ{bullet} \\bullet{asymp} \\asymp{}....and so on.... \\\n\n$${times}$$= Latex symbology examples\n\nIn the \"above link\" to John Forkosh Associates, I used the pre \"enclosed html brackets and \/pre to finish article link even though it had been redone, as a demonstration to illustrate the symbolizations in latex rendering.\n\nClick on Image for a larger size\n\nClick on Image for a larger size\n\nClick on Image for a larger size\n\nAbove, Clifford has laid it out for demonstration within his Sandbox to help others within his bloggery. Hope people use it. I am a little shy when it comes to demonstrating my ignorance, so having my \"own sand box\" would be nice without polluting someone else.\n\nImaging in Latex\nClifford:This is a space, easily accessible from the front page, where you can practice your LaTeX\/MimeTeX commands for writing equations for adding to discussions, etc. You can see several people\u2019s experiments at the earlier post here.\n\n1st generation plotted according to weak hypercharge and weak isospin. Suggestive that the antiparticles are defined rather arbitrarily, and that the structure of idempotents of Clifford algebras (hypercubes) be used to model internal symmetries.\n\nSo having the understanding of what is necessary in the language development of latex, I am still far beneath those who I have linked in terms of that development. But, what ever the case, anything that can help, should not be so far beneath us that we ignore what had been found wanting in those who live the life as a scientist.","date":"2013-05-25 18:01:30","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\": 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\": 0.7617628574371338, \"perplexity\": 1904.1591353603621}, \"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-2013-20\/segments\/1368706009988\/warc\/CC-MAIN-20130516120649-00021-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
Horniki Górne (kaszb. Gòrné Horniczi) – część wsi Nowa Karczma w Polsce położona w województwie pomorskim, w powiecie kościerskim, w gminie Nowa Karczma.
Wieś jest częścią składową sołectwa Nowa Karczma.
W latach 1975–1998 miejscowość administracyjnie należała do województwa gdańskiego.
Nazwy źródłowe miejscowości
Arniki Górne, niem. Ober Hornikau
Przypisy
Zobacz też
Grabowo Kościerskie
Jasiowa Huta
Nowa Karczma (gmina Nowa Karczma) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,443 |
Gradual liberalisation: Efforts are being made to slowly open the country's markets
AlgeriaEconomy
Algeria's trade patterns have long been quite stable, with exports relying overwhelmingly (97%) on hydrocarbons and trade with the EU accounting for roughly half of the total. Bolstered by still-high oil prices into the first half of 2013, Algeria maintains beneficial terms of trade that allow it to run a consistent surplus.
Although there is limited impetus for a change in trade policy, authorities are forging ahead with plans to gradually liberalise trade both on a bilateral basis, with an agreement entering into force with neighbouring Tunisia in 2013, and on a multilateral basis as World Trade Organisation (WTO) accession talks gather pace. A more active trade policy will prove beneficial in taming potential imported inflation as well as in supporting the country's industrialisation strategy.
CONSISTENT SURPLUS: The global economic crisis led to a drop in Algerian exports; however, the country's trade surplus has since recovered, from $7.8bn in 2009 to $18.2bn in 2010, $26.2bn in 2011 and $27.2bn in 2012, before dropping to $8.16bn in the first nine months of 2013, according to the National Centre for Information and Statistics (Centre National de l' Informatique et des Statistiques, CNIS). The current account surplus' value, which reached 8.2% of GDP in 2012, according to the African Development bank (AfDB), fluctuates in line with world oil prices. Trade with the EU grew some 40% in value in the four years to 2012, driven primarily by rising oil prices, as per EU data.
Despite this, hydrocarbons' share of total exports dropped slightly from over 97% in 2012 to 96.09% in the first half of 2013, a function of slightly lower oil prices and higher non-oil and gas exports, which grew a remarkable 66.23% year-on-year (y-o-y) in the first half, albeit from a low base. Raw materials account for 76.15% of non-oil exports, followed by foodstuffs (0.15%) and semi-finished products (0.08%), according to AfDB. The main drivers of non-hydrocarbons growth were oils and affiliated products derived from tar production, which are used in industrial processes – these grew 269.8% y-o-y. The two other leading non-oil exports, ammonia as well as cane and beetroot sugar, remained relatively stable.
Imports have remained largely unchanged in recent years at $47.2bn in 2011 and $46.8bn in 2012, having risen 15.5% from the $40.5bn reported in 2010. Intermediate materials and capital goods used for industrial production dominate the import bill, accounting for 30% and 29% of imports, respectively, in 2012, according to AfDB. Yet in the first half of 2013, the value of imports surged some 17.99% y-o-y, driven by increases of 20.99% intermediate materials, 18.56% in capital goods, 14.96% in food imports and 15.03% in non-food consumer goods, according to CNIS.
STABLE PATTERNS: Algeria has significantly diversified its trading partners over the past two decades and now counts the US as its largest export market, accounting for 16.14% of exports in 2012, according to data from the Ministry of Commerce (MoC). The next three markets are European – Italy with 15.77% of Algerian exports, Spain with 10.23% and France with 8.92% – followed by Canada (7.41%) and the Netherlands (7.15%). On the import front, France remains the largest supplier with 12.83% of Algerian imports. The value of imports from China has ballooned over the last decade, however, rising from roughly $200m to $8bn in the 12 years to 2012, according to the Chinese Embassy in Algiers. Imports from China have continued to grow apace, registering 24% y-o-y growth in 2012 alone, when China supplied 12.56% of Algerian imports. The next four largest sources of imports have remained relatively stable – Italy (9.29%), Spain (8.74%), Germany (5.5%) and Argentina (3.85%). This stable pattern, particularly of exports, is changing due to the ramping up of US shale gas production, however. By February 2013, oil exports to the US dropped to 16,000 barrels per day (bpd), down from roughly 260,000 bpd one year prior.
PROMOTION: The Export Promotion Agency, or Algex, was established in 2004 and has expanded incentives for non-oil exporters in recent years. With roughly $40bn in non-oil imports but a mere $2bn in non-oil exports, the need to rebalance Algeria's non-oil trade position is clear. The Special Export Promotion Fund, operated by the MoC since 1996, supports exporters in their market studies, road-shows and other marketing activities. In 2012 alone it disbursed AD766m (€7.05m) to 317 exporters. The MoC enacted reforms in September 2011 to expand support, by allowing exporters to keep 20% of their earnings in foreign currency (up from 10%), providing for the reimbursement of value-added taxes and duties on exported goods, and establishing a National Shipping Council to improve trade logistics. It also created a new training programme for exporters in cooperation with the Algerian Chamber of Commerce & Industry and the national business school. More broadly, the reforms established an export strategy aimed at developing new markets as diverse as South Africa and Pakistan, as well as expanding support for non-oil exports. The government also enacted import restrictions in July 2009, which require the use of letters of credit for the payment of imports. Additional restrictions were implemented in 2010, when a new tax was imposed on durum wheat imports at prices below domestic Algerian rates (although the national Cereal Agency is exempted from these). According to the WTO, as of 2010, Algeria imposed average import duties of 18.6% – an average of 23.3% for agricultural goods and 17.8% for non-agricultural goods.
TRADE LIBERALISATION: Targeted import duties notwithstanding, Algeria's recent policy has been to slowly move towards liberalisation. As a member of the Greater Arab Free Trade Area (GAFTA) since 2009, which confers tariff-free trade rights amongst all 17 members, Algeria has seen its trade with the group grow, with non-oil exports (mainly sugar and glass) rising 21% to $267m in 2012, representing 12% of its total exports. Yet Algeria continues to import far more from the zone, with imports accounting for 88.3% of total Algeria-GAFTA trade in 2012, according to the CNIS.
Algeria has shown more caution in implementing the European Neighbourhood Policy (ENP). Although it signed an association agreement with the EU in 2008 that aimed at creating a free trade zone for industrial goods by 2017, with removal of tariffs on 82 products due to be completed by September 2012, it decided to delay full implementation until 2020. "This revision was requested because the association agreement did not have the expected impact in terms of promoting non-hydrocarbons exports," Trade Minister Mustapha Benbada told OBG. "The tariff advantages were not effectively used to diversify the national economy and ensure mutual profit." Preferential market access rights conferred to Algeria under the Generalised System of Preferences (GSP), which abolishes duties on Algerian exports, will be removed from January 2014 and replaced by the ENP for Maghreb countries.
Although Algeria has yet to sign on to the policy, its hydrocarbons exports will continue to be exempted. While more marginal at only $1bn in 2012, Algeria's trade with neighbouring Tunisia will receive a boost from the enactment of a bilateral trade agreement in August 2013. The deal confers domestic tax treatment on goods produced in the other country and removes between 40% and 100% of Customs duties on trade in industrial goods. Agricultural and meat products, as well as all foodstuffs, are fully exempt from such duties.
WTO TALKS: These piecemeal accords represent encouraging signs of gradual liberalisation, yet Algeria's global trade relations remain tied to negotiations on WTO accession, originally launched in 1995. After a series of fits and starts, Algeria placed renewed impetus on talks by holding the 11th round in April 2013, proposing more significant reforms than in the past, according to the WTO. Additional reforms will be needed, however, in a broad set of areas ranging from the foreign investment regime, the state's economic role and competition policy, and foreign-exchange and fiscal policies, according to the minister of trade. While domestic lobbies have agitated against accession, global bodies like the IMF have called for a swift conclusion of negotiations in order to boost Algeria's productivity growth. "The need to increase productivity is the biggest argument in favour of accession," Benbada told OBG. "This is a powerful factor in the growth that will strengthen the competitiveness of the national economy in the post-oil era." This is proving increasingly pressing given the IMF's projections of a narrowing trade surplus in the four years to 2017, based on forecasts of a declining oil price spurred by greater US self-sufficiency in oil and gas. Indeed, the IMF estimates that the trade surplus will decline from $25.9bn in 2013 to $14.6bn in 2017 – with exports dropping 12.3% in value to $62.7bn in four years, while imports are expected to grow 5.3% to $48bn in the same span.
While pursuing trade liberalisation and an expansion in non-oil exports has never been straightforward, given major windfall oil revenues, the authorities are conscious of the need to increase the country's integration with the global economy in order to support other development objectives. Expanding such exports will be key to improving Algeria's competitiveness, productivity and eventually its job-creating potential.
The Report: Algeria 2013
Economy chapter from The Report: Algeria 2013
OBG talks to Mustapha Benbada, Minister of Trade Next article from this chapter and report
OBG talks to Mohamed Seghir Babes, President, National Economic and Social Council (CNES)
A key resource: Hydrocarbons production continues to play a crucial role in the economy
OBG talks to Karim Djoudi, Minister of Finance
OBG talks to Mustapha Benbada, Minister of Trade
OBG talks to Réda Hamiani, President, Algerian Business Leaders' Forum
Consolidation, not austerity: Prudent measures are being taken to help restrain government spending
A helping hand: Many staple products receive either direct or indirect government subsidies
Christine Lagarde, Managing Director, IMF, on ensuring continued stability
This article is from the Economy chapter of The Report: Algeria 2013. Explore other chapters from this report.
Béjaïa
Featured Sectors in Algeria:
Algeria Financial Services
Algeria Energy
Algeria Economy
Algeria Industry
Algeria Construction | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,960 |
\section{Introduction} \label{sec:intro}
In the low density environment of the solar corona, the magnetic field governs both the atmospheric structure and the energetics of active solar phenomena like flares, coronal mass ejections and the solar wind. Two promising techniques available for remotely measuring the coronal magnetic field are spectropolarimetric measurements of emission lines in the optical part of the spectrum and radio observations from coronal regions with different plasma frequencies \citep{Casini17}; however, each face observational and interpretation challenges.
The brightest observable visible/infrared (Vis/IR) coronal emission lines are magnetic dipole (M1) lines \citep{Judge98, DelZanna18a}. Photoexcitation of these lines by anisotropic disc radiation leads to scattering induced linear polarization with amplitudes of ${\sim}10^{-2}$ to $10^{-1}$ of the total line intensity \citep[e.g.,][]{Arnaud87, Tomczyk08}). The presence of the magnetic field modifies the linear polarization through the saturated Hanle effect (\textit{i.e.}, its amplitude is not sensitive to the magnetic field intensity) and induces a small circularly polarized signal through the longitudinal Zeeman effect with relative amplitudes of ${\sim}10^{-4}$ to $10^{-3}$ for a 10 G magnetic field \citep{Harvey69, Lin00, Lin04}. Achieving high sensitivity measurements of these signals is challenged by the large ${\sim}10^{5}$ brightness contrast between the solar disc and coronal emission in the Vis/IR \citep{Arnaud82, Arnaud87, Kuhn96, Madsen19} and requires mitigation of scattered light from the sky and telescope optics. Observations at longer wavelengths are typically more sensitive because the scattered light performance of telescope optics improves and the Zeeman splitting increases faster than the thermal line width \citep{Penn14}. The upcoming 4m \textit{Daniel K. Inouye Solar Telescope} (DKIST) will have two first light spectropolarimetric instruments targeting multiple IR emission lines built to address these challenges: DL-NIRSP and Cryo-NIRSP \citep{Rimmele15}.
Key theoretical advances in understanding the formation of polarized M1 lines in the corona have enabled new diagnostic capabilities based on synthesizing polarimetric signals through the optically thin corona \citep{Charvin65,SahalBrechot77,House77,Casini99,Casini00}. Due in part to the current sparsity of circular polarization detections, most progress has been made by comparing observed linear polarized signals of a single M1 line to those synthesized through forward models \citep{Judge06,Dove11,Rachmeler13,Dalmasse19} or by using time-based tomographic methods to invert the global coronal magnetic structure \citep{Kramar06, Kramar13}, again primarily using a single M1 line. These techniques necessarily consider the emission contributed by all the plasma along each line of sight. Work by \cite{Liu08} succeeded in comparing measured circular polarization signals with those synthesized through an active region.
In contrast to the above approaches, here we focus on using full Stokes observations of multiple M1 lines to invert the full vector magnetic field for an isolated region with uniform magnetohydrodynamic properties: density, temperature and magnetic field. This region either dominates the total emission along the line of sight or can potentially be separated from the foreground/background emission with subtraction methods. Good candidate targets are coronal loops that appear in contrast against the background corona in ultraviolet \citep{Brooks12,Aschwanden13,Xie17} and IR \citep{Tomczyk08}. \citet{Plowman14} demonstrated that, under this ``single point approximation'', using a combination of full Stokes observations from the \ion{Fe}{13} 10747 \AA{} and 10798 \AA{} lines, all three components of the magnetic field could be analytically calculated without explicit knowledge of the thermodynamic parameters that influence the amplitude of the linear polarization.
In this work we examine the single point inversion method further by generalizing the approach of \citet{Plowman14} to consider, in principle, any combination of M1 lines. The analytical formalism we derive in Section \ref{analytic_solution} for two line observations reveals that only specific line combinations may be used for the inversion due to fundamental quantum mechanical degeneracies in the polarized emission coefficients. As a result, we learn that the \ion{Fe}{13} line pair used by \citet{Plowman14} may not be adequate for single point inversions if LS coupling is valid for these lines. In Section \ref{sec:noise} we propose combining the \ion{Fe}{13} 10747 \AA{} and \ion{Si}{10} 14301 \AA{} coronal lines to implement the single point inversion. We also evaluate the impact photon noise has on this line combination when attempting to infer the magnetic field vector. In Section \ref{discussion} we discuss ambiguities inherent in the present inversion approach and as well as other related diagnostics.
\section{Analytic inverse solution}
\label{analytic_solution}
Starting from equations 35(a-c) in \citet{Casini99,Casini00}, and making use of the geometric tensors for magnetic dipole radiation given by their equations 39(a-d), the polarized emission coefficients ($\epsilon_{i}$) for the Stokes parameters ($i=I,Q,U,V$) of an M1 line can be written as a function of wavelength (as opposed to angular frequency) as:
\begin{align}
\epsilon_I(\lambda) &= C\left[1 + \frac{3}{2\sqrt{2}}D\sigma^2_0\left(\cos^2{\Theta_B} - \frac{1}{3}\right)\right]\phi(\lambda-\lambda_0)\label{eqI0}\\
\epsilon_Q(\lambda) &= \frac{3}{2\sqrt{2}}CD\sigma^2_0\sin^2{\Theta_B}\cos{(2\gamma_B)}\phi(\lambda-\lambda_0) \label{eqQ0}\\
\epsilon_U(\lambda) &= -\frac{3}{2\sqrt{2}}CD\sigma^2_0\sin^2{\Theta_B}\sin{(2\gamma_B)}\phi(\lambda-\lambda_0) \label{eqU0}\\
\epsilon_V(\lambda) &= -C\frac{\lambda^2}{c}\nu_L\cos{\Theta_B}\left[\bar{g}+E\sigma^2_0\right]\phi'(\lambda-\lambda_0) \label{eqV0}
\end{align}
where $C$ (\textit{i.e.}, $C_{JJ_{0}}$ in the original notation) is the emission coefficient (without stimulated emission) in the unpolarized case and depends on the total upper level population and the Einstein A coefficient for the transition. $D=D_{JJ_{0}}$, $E=E_{JJ_{0}}$ and $\bar{g}=\bar{g}_{JJ_{0}}$ in the original notation are atomic parameters that depend on the quantum numbers of the upper and lower levels of the transition, $J$ and $J_0$ respectively. $\Theta_B$ is the inclination angle of the magnetic field with respect to the line of sight while $\gamma_B$ is the azimuth angle of the reference direction of linear polarization (r.d.l.p) measured from the projected magnetic field direction (see Figure \ref{fig:coordinate_system} as well as Figure 5 in \cite{Casini99}). The Larmor frequency $\nu_{L}$ is proportional to the magnetic field strength B. $\phi(\lambda-\lambda_0)$ refers to the Doppler dominated normalized ($\int \phi{\ }d\lambda = 1$) spectral profile centered at $\lambda_0$ while $\phi'(\lambda-\lambda_0)$ is its derivative with respect to wavelength.
The scalar quantity $\sigma^2_0$ refers to the fractional atomic alignment of the upper level of the transition which describes the imbalance of the magnetic sublevel populations for different absolute values of the magnetic quantum number. It is equal to $\rho^2_0/\rho^0_0$ where the tensor $\rho^{K}_{Q}$ is the irreducible spherical statistical tensor representation of the atomic density matrix. $\sigma^2_0$ can be positive or negative and is generally a function of the exciting photospheric radiation anisotropy (which varies with height), the plasma density and temperature, and the local inclination angle of the magnetic field in the local radial frame of reference $\theta_B$. For the M1 lines of interest, $\sigma^2_0$ can often be well approximated by a positive definite factor, $k_{J}(T_{e},n_{e},h),$ dependent on the electron temperature, electron density, and height multiplied by a functional dependence on $\theta_B$ given by $3 \cos^2{\theta_B} - 1$ \citep{Judge07}, which gives rise to the Van Vleck ambiguity discussed in Section~\ref{ambiguities}. Note that $\sigma^2_0$ does not depend on the magnetic field intensity.
In principle, given proper constraints on the local plasma properties and the radiation field, as well as numerical calculations of $k_{J}$, full Stokes observations of a single M1 line can be used to derive the full magnetic field vector, within certain limitations and within the single point approximation as discussed in \cite{Judge07}. In absence of such extra constraints, a full set of Stokes observations of a single M1 line provide four measurements for five unknowns (\textit{i.e.}, $C,\sigma^2_0,\gamma_B, \Theta_{B},B$). $\gamma_B$ can be derived using the ratio of U and Q; but the remaining variables remain coupled. With some limitations, to be discussed in Section~\ref{blos}, the projected magnetic field intensity along the line of sight ($B_{LOS}\equiv B\cos\Theta_B$) can be derived from polarimetric observations of a single line. Likewise, the value of $C$ can be determined from I by neglecting the atomic alignment contribution; but $\sigma^2_0$ and $\Theta_B$ cannot be directly determined.
\begin{figure}
\centering
\includegraphics[width=3.3in]{coordinate_system_6.pdf}
\caption{Angular coordinates between three vectors passing through a point of emission in the corona: the outward radial direction (\textbf{R}), magnetic field (\textbf{B}) and line of sight (\textbf{k}). The projected magnetic field in a plane perpendicular to the line of sight is characterized by the azimuth angle $\gamma_B$ measured clockwise towards the reference direction for linear polarization (r.d.l.p.) with a 180\deg ambiguity since the linear polarization direction is not oriented. Figure adapted from Figure 5 in \citet{Casini99}.}
\label{fig:coordinate_system}
\end{figure}
\citet{Plowman14} showed that with full Stokes measurements of two M1 lines, in particular the \ion{Fe}{13} 10747, 10798 \AA{} line pair, it is possible to obtain explicit analytic formulas for all the unknown parameters in equations \ref{eqI0}-\ref{eqV0} under the single point assumption. An additional spectral line provides four more measured quantities but only two additional unknowns, as the magnetic field properties are assumed to be the same. \citeauthor{Plowman14} focused only on the \ion{Fe}{13} lines, using analytic formulas for the D and E parameters from Equation 38(a-b) in \citet{Casini99} before solving for the unknown parameters.
To extend this approach to other combinations of M1 emission lines, here we retain the symbolic form for the parameters D, E. Under the single point approximation, the line is strongly dominated by emission from a single region in space so the measured polarized Stokes profiles $S_i(\lambda)$ can be approximated as:
\begin{align}
\begin{array}{ll}
S_i(\lambda) = \int_0^{\infty}\epsilon_i(\lambda)ds \approx \epsilon_i(\lambda)\Delta s & (i=I,Q,U,V)
\end{array}
\end{align}
where the integral is over emission along the line of sight and $\Delta s$ is the length of the homogeneous single point region. From the measured dispersed Stokes profiles the wavelength-independent Stokes amplitudes for the two lines can then be derived as:
\begin{align}
I &= \frac{S_I}{\phi(\lambda-\lambda_{0})} \approx \frac{\epsilon_I\Delta s}{\phi(\lambda-\lambda_{0})} \nonumber \\
&\approx I_0\left[1 + \frac{3}{2\sqrt{2}}D\sigma^2_0\left(\cos^2{\Theta_B} - \frac{1}{3}\right)\right] \label{eqI}\\
L &= \frac{\sqrt{S_Q^2 + S_U^2}}{\phi(\lambda-\lambda_{0})} \approx \frac{\sqrt{\epsilon_Q^2 + \epsilon_U^2}\Delta s}{\phi(\lambda-\lambda_{0})} \nonumber \\
&\approx \pm \frac{3}{2\sqrt{2}} D I_0 \sigma^2_0 \sin^2{\Theta_B} \label{eqL}\\
V &= \frac{cS_V}{\lambda^2\phi'(\lambda-\lambda_{0})} \approx \frac{c\epsilon_V\Delta s}{\lambda^2\phi'(\lambda-\lambda_{0})} \nonumber \\
&\approx -\frac{1}{\sqrt{2}}I_0\nu_L\cos{\Theta_B}\left[\left(\sqrt{2}+D\sigma^2_0\right)\bar{g}+FD\sigma^2_0\right] \label{eqV}
\end{align}
where $I_0 \equiv C\Delta s$ is a constant with units of intensity. We combine the Q and U Stokes parameters into a single parameter L, which is the total linear polarization. We keep both signs of the quadratic sum because, as already stated, the sign of the atomic alignment $\sigma^2_0$ can be either positive or negative while the measured quantity L is always positive. We also introduce the parameter F defined as:
\begin{align}
F &\equiv \sqrt{2}\frac{E}{D} - \bar{g} \nonumber\\
&= \frac{3}{4}[J(J+1) - J_0(J_0+1) -2](g_1 - g_0) \label{eqF}
\end{align}
where $g_1$ and $g_0$ are the Land\'e factors for the upper and lower levels in the transition. The analytic expression for F can be derived from Equations 36b-c in \citet{Casini99} for $D$ and $E$ by expanding the Wigner 6-j and 9-j coefficients and is valid for all M1 transitions with $\Delta J=0,\pm1$. This parameter is introduced in hindsight since it will be shown to play an important role in discriminating between lines that contain non-degenerate information.
The key to the two-line inversion involves solving for the angle $\Theta_B$ that gives the same value for B assuming lines are emitted from the same spatial point with the same magnetic field properties. We first write the magnetic field strength B as a function of $\Theta_B$ by substituting equations \ref{eqI} and \ref{eqL} into equation \ref{eqV} and using the expression for the Larmor frequency (\textit{i.e.}, $\nu_L=(\mu_B/h)B$):
\begin{align}
B = \frac{h}{\mu_B}\frac{-V}{\left[(I \pm L)\bar{g} \pm \frac{2}{3}F\frac{L}{\sin^2{\Theta_B}}\right]\cos{\Theta_B}} \label{eqB_num1}
\end{align}
where $h$ and the $\mu_B$ are Planck's constant and the Bohr magneton respectively. Using this equation and subscripts 1 and 2 for the two lines, we assume $B_{1} \equiv B_{2}$ and $\Theta_{B1} \equiv \Theta_{B2}$ and then solve for $\sin^{2}\Theta_{B}$. The other unknown parameters then follow from Equations~\ref{eqI},~\ref{eqL},and~\ref{eqB_num1}. To summarize, the analytical inversion equations are as follows:
\begin{align}
\sin^{2}\Theta_{B} &= \pm \frac{2}{3}\frac{F_1L_1V_2 - F_2L_2V_1}{\bar{g}_2V_1(I_2 \pm L_2) - \bar{g}_1V_2(I_1 \pm L_1)}, \label{eqsin}\\[5pt]
I_{0i} &= I_i \pm L_i\left(1 - \frac{2}{3\sin^2{\Theta_B}}\right), \\[5pt]
\sigma^{2}_{0i} &= \frac{\pm L_i}{\sqrt{\frac{9}{8}} I_{0i} D_i \sin^{2}\Theta_{B}}, \\[5pt]
B &= \frac{h}{\mu_B}\frac{-V_{i}}{\left[(I_i \pm L_i)\bar{g}_i \pm \frac{2}{3}F_i\frac{L_i}{\sin^{2}{\Theta_{B}}}\right]\cos{\Theta_{B}}},\label{eqB}\\
\gamma_B &= \left\{
\begin{array}{ll}
-\frac{1}{2}\arctan\left(\frac{U_i}{Q_i}\right) + \{0,\pi\}, & \sigma^2_{0i} > 0 \\[5pt]
-\frac{1}{2}\arctan\left(\frac{U_i}{Q_i}\right) + \{\frac{\pi}{2}, \frac{3\pi}{2}\}, & \sigma^2_{0i} < 0 \\
\end{array}
\right. \label{eqgamma}
\end{align}
where the arctangent function must account for the signs of the arguments to determine the quadrant for the result (usually implemented numerically as the arctan2 function). Since M1 lines are in the saturated Hanle regime, the magnetic field projection ($\gamma_B$) is parallel or perpendicular to the direction of linear polarization depending on sign of the alignment and the field inclination relative to the outward radial direction $\theta_B$ (Van Vleck effect, see discussion in Section \ref{ambiguities}). In addition, the reference direction of linear polarization has a $180^{\circ}$ ambiguity, which introduces the additional solutions indicated in the curly brackets of Equation~\ref{eqgamma}. Consequently, given two measured Stokes vectors this inversion method produces up to four degenerate solutions for the magnetic field (discussed further in Sections~\ref{sec:noise} and ~\ref{ambiguities}). The sign for $\cos\Theta_B$ is chosen such that B is positive in equation \ref{eqB} since physically the magnetic field strength is a positive quantity.
An important result can be seen from equation \ref{eqsin}: emission lines with $F=0$ contain degenerate information about the magnetic field such that no unique solution for $\Theta_B$ can be deduced from the two Stokes vectors. To apply this type of inversion emission lines must be selected such that at least one line has $F\ne0$. From Equation~\ref{eqF}, F is zero when the upper and lower levels have equal Land\'e g-factors ($g_1=g_0$) or for $J=1\rightarrow0$ transitions.
Table~\ref{tab:lines} reports the values of the D and F atomic parameters for a list of candidate visible and infrared M1 lines identified by \citet{Judge01} as having the best coronagraphic potential. The values are calculated assuming LS coupling such that the Land\'e factors for each level are given by
\begin{align}
g = \frac{3}{2} + \frac{S(S+1) - L(L+1)}{2J(J+1)},\label{eq:LScoupling}
\end{align}
where S and L are the total spin and orbital angular momenta. We find that only three of the candidate lines have $F\ne0$---\ion{Fe}{10} 5303 \AA{}, \ion{Si}{10} 14301 \AA, and \ion{Mg}{8} 30285 \AA. The \ion{Fe}{10} 6375 \AA{} line can only be circularly polarized because the upper level has $J=1/2$ and it therefore cannot be atomically aligned. Transitions with an upper level that is not polarizable have $D=0$, $F=0$ and no alignment correction is needed to measure an unbiased value for the line of sight magnetic field (see discussion in Section \ref{blos}).
Based on these results, the \ion{Fe}{13} 10747 \AA{} and 10798 \AA{} lines considered by \citet{Plowman14} each have $F=0$ and thus contain degenerate information. Consequently, this line pair does not allow one to invert for the magnetic field vector. The discrepancy between his results and ours arises due to the use of Equation 38b from \citet{Casini99} which contains a small algebraic error leading to a spurious value for the coefficient $E$ for the \ion{Fe}{13} 10798 \AA{} line.\footnote{We have confirmed this error with the authors of \cite{Casini99} and \cite{Plowman14}.} That said, the information in these line pairs is only degenerate if LS coupling holds for the energy levels. This may not be case for the \ion{Fe}{13} 10747, 10798 \AA{} pair because the separation in the energy levels for the two transitions does not obey Land\'e's interval rule, which states that the splitting between two fine structure levels of the same term is proportional to the larger J value. The ratio of energies between the $^3P_2 \rightarrow\ ^3P_1$ and $^3P_1 \rightarrow\ ^3P_0$ transitions in the ground term of \ion{Fe}{13} is 9258.6/9303.1 $\sim1$ instead of 2 as would be expected from Land\'e's interval rule. It may be that the \ion{Fe}{13} line pair does contain non-degenerate information but confirmation of the Land\'e factors for the levels is needed before the line can be used as a diagnostic for the magnetic field.
Recent laboratory experiments by \citet{Hensel16} and \citet{Bekker18}, using electron beam ion traps, succeeded in measuring the Land\'e g-factors for visible transitions from highly ionized iron, specifically \ion{Fe}{14} 5303 \AA, \ion{Fe}{10} 6475 \AA{} and \ion{Fe}{11} 7892 \AA. The g-factors measured for \ion{Fe}{11} 7892 \AA{} indicate significant departures from LS coupling; however, the g-factors for the two levels of the transition are equal within the experimental uncertainties which still means $F=0$ for this line. Extending such laboratory measurements to the infrared transitions of iron, as well as the other species in Table \ref{tab:lines}, is necessary to ensure the accuracy of coronal magnetic field measurements.
\begin{deluxetable}{lr|ccccccccccccc}
\tablecaption{D and F atomic parameters for candidate visible and IR magnetic dipole coronal lines selected by \cite{Judge01} assuming LS coupling for the Land\'e g-factors. \label{tab:lines}}
\tablehead{
\colhead{Ion} & \colhead{$\lambda_0$(air)[\AA]} & \colhead{Transition (u$\rightarrow$l)} &
\colhead{D$_{JJ_0}$} & \colhead{F$_{JJ_0}$}
}
\startdata
\ion{Fe}{14} & 5303\tablenotemark{a} & $^2P_{3/2} \rightarrow\ ^2P_{1/2}$ & 0.71 & 0.5\\
\ion{Fe}{10} & 6375\tablenotemark{a} & $^2P_{1/2} \rightarrow\ ^2P_{3/2}$ & 0.0 & -\tablenotemark{e}\\
\ion{Fe}{11} & 7892\tablenotemark{a} & $^3P_1 \rightarrow\ ^3P_2$ & 0.11 & 0.0 \\
\ion{Fe}{13} & 10746\tablenotemark{b} & $^3P_1 \rightarrow\ ^3P_0$ & 1.0 & 0.0\\
\ion{Fe}{13} & 10798\tablenotemark{c} & $^3P_2 \rightarrow\ ^3P_1$ & 0.6 & 0.0\\
\ion{S}{9} & 12524\tablenotemark{d} & $^3P_1 \rightarrow\ ^3P_2$ & 0.1 & 0.0 \\
\ion{Si}{10} & 14301\tablenotemark{b} & $^2P_{3/2} \rightarrow\ ^2P_{1/2}$ & 0.71 & 0.5\\
\ion{S}{11} & 19201\tablenotemark{d} & $^3P_1 \rightarrow\ ^3P_0$ & 1.0 & 0.0 \\
\ion{Si}{7} & 24826\tablenotemark{d} & $^3P_2 \rightarrow\ ^3P_1$ & 0.1 & 0.0 \\
\ion{Mg}{8} & 30285\tablenotemark{d} & $^2P_{3/2} \rightarrow\ ^2P_{1/2}$ & 0.71 & 0.5\\
\ion{Si}{9} & 39343\tablenotemark{c} & $^3P_1 \rightarrow\ ^3P_0$ & 1.0 & 0.0 \\
\ion{Mg}{7} & 55033\tablenotemark{d} & $^3P_2 \rightarrow\ ^3P_1$ & 0.6 & 0.0\\
\ion{Mg}{7} & 90090\tablenotemark{d} & $^3P_1 \rightarrow\ ^3P_0$ & 1.0 & 0.0
\enddata
\tablenotetext{a}{Laboratory measurements of g-factors available from \citet{Bekker18} but not used here.}
\tablenotetext{b}{\citet{Dima19a}}
\tablenotetext{c}{\citet{DelZanna18a}}
\tablenotetext{d}{CHIANTI version 8 \citep{DelZanna15}}
\tablenotetext{e}{Not defined since $D{=}0$ and $E{=}0$}
\end{deluxetable}
\section{The influence of observational noise \\ on inversions of \texorpdfstring{\ion{Fe}{13} and \ion{Si}{10}}{Fe XIII and Si X }}\label{sec:noise}
Coronal measurements are background-limited \citep{Penn04}, and current observatories like COMP \citep{Tomczyk08} can only routinely achieve noise levels at the $10^{-3}$ noise level, while upcoming facilities like DKIST \citep{Rimmele15} and COSMO \citep{Tomczyk16} will more routinely achieve observations at the $10^{-4}$ noise level and higher. Following \citet{Plowman14}, here we consider the influence of observational noise on the inverted quantities using a line combination that meets the criteria set forth above.
We propose the \ion{Fe}{13} 10747 \AA{} ($F=0$) and \ion{Si}{10} 14301 \AA{} ($F=0.5$) line combination as a promising alternative to the \ion{Fe}{13} line pair. Of the lines with $F\ne0$ identified, the \ion{Si}{10} 14301 \AA{} has perhaps the best potential for full Stokes measurements on account of its brightness and heightened Zeeman sensitivity due to its longer wavelength \citep{Judge01}. Meanwhile \ion{Fe}{13} 10747 \AA{} has already been demonstrated to be a line with good potential for coronal magnetometry \citep{Lin00,Lin04}. Both lines are photoexcited from the lower level of the ground state and have large values for D, which in turn leads to larger atomic alignments and larger linear polarization. While the temperatures of peak ionization fraction for \ion{Fe}{13} and \ion{Si}{10} are different (${\sim}$1.7 MK and 1.4 MK, respectively), the contribution functions do overlap significantly such that structures like coronal loops between 1.5 and 1.6 MK will have measurable emission in both lines. Furthermore, both lines have now been measured simultaneously during eclipses and using ground-based coronagraphic linear polarimetry \citep{Dima18, Dima19a}. One drawback to using line pairs from different ionized species is the increased likelihood that emission in each line may not originate in the same coronal region. However, as discussed in Section \ref{sec:implementation} analyzing the polarization angle for each line offers an important check on whether the emission is co-spatial.
Here we investigate the effect of noise on inversions of this line combination by performing a Monte Carlo experiment based on inverting synthetic observations of the two lines. Using a modified version of the polarized coronal line emission code \citep[CLE,][]{Judge01b}, which numerically calculates the atomic alignment and level populations for statistical equilibrium of a multi-level atom including photoexcitation and electron collisions, we generated dispersed full Stokes line profiles for the \ion{Fe}{13} 10747 \AA{} and \ion{Si}{10} 14301 \AA{} lines sampled at 0.035 \AA. Example line profiles of the \ion{Fe}{13} 10747 \AA{} line are shown in Figure \ref{fig:line_fit_noise} for normally distributed noise in each pixel with standard deviations of $10^{-3}I_{peak}$ and $10^{-4}I_{peak}$. The \ion{Si}{10} profiles are comparable and not shown. The spectra shown are for emission located in the plane of the sky passing through the center of the Sun and at a height of $0.1$ \Rsun{} above the photosphere. The reference direction for linear polarization is perpendicular to the plane containing the line-of-sight and the radial direction. The input magnetic field has local radial coordinates (B, $\theta_B$, $\phi_B$) = (25 G, 90\deg, 40\deg) and line of sight coordinates (B, $\Theta_B$, $\gamma_B$) = (25 G, 40\deg, 0\deg). The electron density is set to $n_e = 10^{8.5}$ cm$^{-3}$ and the temperature is $\sim1.55$ MK. As we are using relative noise values, we do not need to set a value for the total emission measure (or length of plasma along line of sight); however, the general scenario here is that of an over-dense closed coronal loop observed near its apex above the solar limb \citep[see, \textit{e.g.}][]{Aschwanden13}.
\begin{figure}
\centering
\includegraphics[width=3.3in]{line_fit_noise_90.pdf}
\caption{Synthetic noisy line profiles (black) and fits (red) for the \ion{Fe}{13} 10747 \AA{} line. The spectra sampling is 0.035 \AA{} and normally distributed random photon noise is added to each pixel with the standard deviation indicated above each column. The emission region is located at a height of 0.1 \Rsun{} and the scattering angle is $\theta_B = 90^{\circ}$. The input magnetic field has line of sight coordinates (B, $\Theta_B$, $\gamma_B$) = (25 G, 40\deg, 0\deg).}
\label{fig:line_fit_noise}
\end{figure}
Like \cite{Plowman14} we test the single point inversion for different inclination angles relative to the line of sight ($\Theta_B$) for constant values of the atomic alignments ($\sigma^2_0$). To achieve this we fix the strength of the field B and $\theta_{B}$ while varying the azimuth $\phi_B$ in the plane perpendicular to the local radial direction (see Figure \ref{fig:coordinate_system}), which in turn varies $\Theta_B$. Physically this is akin to changing the orientation of a coronal loop from an edge-on ($\Theta_{B} = 0^{\circ}$) to a face-on ($\Theta_{B} = 90^{\circ}$) observational geometry. This approach, in addition to fixing $B \equiv 25$ G, is comparable to that of \cite{Plowman14}. Note, however, that the errors derived will depend upon the magnetic field intensity and the atomic alignments; here only one representative example is considered.
For values of $\Theta^{in}_B$ ranging from $0^{\circ}$ to $90^{\circ}$, where the added superscript indicates ``input'' values, we produce a sample of 500 profiles with different realization of random noise. For each profile, we determine the Stokes amplitude values of I, L, and V in Equations~\ref{eqI}-\ref{eqV} by fitting in a least-squares sense for the normalized profile $\phi(\lambda-\lambda_{0})$ and its derivative $\phi'(\lambda-\lambda_{0})$. The inverted parameters at each noise instance are then obtained from Equations \ref{eqsin}-\ref{eqgamma} and the solutions for each parameter are binned into histograms at each input $\Theta^{in}_B$. $I_{01}$,$\sigma^2_{01}$ and $I_{02}$,$\sigma^2_{02}$ refer to the \ion{Fe}{13} and \ion{Si}{10} lines, respectively. The input values for alignments are $\sigma^2_{01}=-0.021$ and $\sigma^2_{02}=-0.012$. The normalized histograms for the negative and positive solution branches of Equations \ref{eqsin}-\ref{eqgamma} are shown in Figures \ref{fig:negative_sol} and \ref{fig:positive_sol}. Note that $\gamma_B$ and $\gamma_B + \pi$ are also solutions in either case.
In this particular example the input atomic alignments are negative and correct solutions for the parameters are recovered by the negative solution branch (Figure \ref{fig:negative_sol}). The positive solution branch shown in Figure \ref{fig:positive_sol} recovers formally consistent solutions (i.e. solutions with $\mid\cos\Theta_B\mid \leq 1$) for $\Theta^{in}_B > 17^{\circ}$ while for $\Theta^{in}_B<17^{\circ}$ the inversions produce inconsistent solutions with $\cos \Theta_B > 1$. In general the range of $\Theta_B$ for which the number of degenerate solutions decreases is a function of both the lines analyzed and the atomic alignments as discussed in Section \ref{ambiguities}.
\begin{figure*}
\centering
\includegraphics[width=6.5in]{multiple_pars_90_m.pdf}
\caption{Normalized histograms for the parameters calculated using equations \ref{eqsin} - \ref{eqgamma} choosing the negative solution branch for different levels of noise. Each parameter is shown as a function of the input inclination with respect to the line of sight ($\Theta^{in}_B$). Since the original solution had a local inclination angle $\theta_B = 90^{\circ}$ which is larger than the Van Vleck angle the input atomic alignment is negative and this solution branch accurately recovers the input parameters in the noise-free case.}
\label{fig:negative_sol}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=6.5in]{multiple_pars_90_p.pdf}
\caption{Same as figure \ref{fig:negative_sol} but using the positive solution branch. At very low inclination angles ($\Theta_B \lesssim 17^{\circ}$) this solution branch gives inconsistent solutions ($\cos\Theta_B > 1$) effectively solving the $90^{\circ}$ ambiguity. As discussed in the text regions where this happens depend on both the lines used and the physical properties of the emission region.
It is also noticeable that at larger inclination angles the recovered atomic alignments $\sigma^2_{0i}$ are very large. Atomic modeling can be used to rule out solutions with such large atomic alignments based on the scattering geometry.}
\label{fig:positive_sol}
\end{figure*}
We find similar noise effects on the inverted parameters using the \ion{Fe}{13}, \ion{Si}{10} line combination to results discussed by \citet{Plowman14}. Noise levels of $\sim10^{-4}I_{peak}$ ($\sim10^{-5}I_{peak}$) are needed to recover input magnetic field inclinations $\Theta^{in}_B < 36^{\circ}$ ( $\Theta^{in}_B < 66^{\circ}$). Parameters like the atomic alignments and the line intensities are recovered well for noise levels at the $10^{-4}I_{peak}$ levels. The line of sight magnetic field, $B_{LOS}$, and projected magnetic field angle, $\gamma_B$, are particular robust to noise.
In general the full magnetic field vector can be recovered for field inclinations within some limiting angle to the line of sight, $\Theta_B < \Theta^{lim}_B$. The value $\Theta^{lim}_B$ depends on the quantum mechanical properties of the lines used in the inversion, the atomic alignments and the observation noise levels. In the example considered here with a moderately strong field $B=25$ G and small values for the atomic alignments, $\Theta^{lim}_B \approx 36^{\circ}$ for $\sim10^{-4}I_{peak}$ noise levels. Larger noise levels, smaller values for B and smaller atomic alignments all lead to a decrease in $\Theta^{lim}_B$ and a shrinking of the range of invertible solutions.
In agreement with \citet{Plowman14}, we note that the inverted $\Theta_B$ value is biased towards $0^{\circ}$ for larger relative noise and for more transverse input fields directions. \citeauthor{Plowman14} alleviated this bias by applying a Bayesian MCMC method and appropriate priors to an observation and mapping the posterior probability distribution at each input value of $\Theta_B$. Such a method could be applied to any line pair assuming, as demonstrated, at least one of the lines has $F\ne0$.
\section{Discussion}\label{discussion}
\subsection{Ambiguous Solutions}\label{ambiguities}
In the most general case equations \ref{eqsin}-\ref{eqgamma} return four consistent solutions ($\mid\cos\Theta_B\mid \leq 1$) for the inverted set of parameters. Two of the solutions have projected directions $\gamma_B$ at 90\deg{} to each other and map to the classical Van Vleck ambiguity. However, it is notable from looking at Figures \ref{fig:negative_sol} and \ref{fig:positive_sol} that the ambiguity extends to all the inverted parameters. Thus solving this ambiguity may be accomplished by imposing physical limits on any of the solution parameters. The other two ambiguous solutions have parameters identical to the first two except the projected angles $\gamma_B$ are different by 180\deg{} (see example in Figure \ref{fig:all_solutions}).
The number of solutions depends on both the intrinsic quantum mechanical properties of the transitions through the $D$, $F$ and $\bar{g}$ parameters and on the atomic alignments $\sigma^2_{0i}$ which are functions of the extrinsic thermodynamic properties of the plasma and radiation anisotropy. An example of this is discussed in Section \ref{sec:noise}, where the number of solutions decreases to only two (180\deg{} ambiguity) when $\Theta_B\lesssim 17^{\circ}$. When the emission lines used in the inversion have equal $D$ and $F$ values (e.g. \ion{Fe}{14} 5303 \AA{} and \ion{Si}{10} 14301 \AA{} or \ion{Mg}{8} 30285 \AA) the number of degenerate solutions, in the noise free case, reduces to two for all values of $\Theta_B$ and atomic alignments.
While the multi-line method recovers (at most) four possible degenerate solutions for the vector magnetic field and atomic alignment, it does not provide direct knowledge of the scattering angle $\theta_{obs}$, the location of the emission along the line of sight, nor the magnetic field direction in a coordinate frame local to the Sun. The field direction in the solar frame can be calculated from the inferred (degenerate) LOS-frame field vector(s) for a given scattering angle using spherical trigonometry (see equations 42,44 in \citet{Casini99}). With knowledge of the projected height above the solar surface and the inferred values of atomic alignment, numerical calculations (as discussed above) can determine a range of temperature and densities required to be consistent with the observations at a given scattering angle. Consequently, independent observations of the temperature and density can constrain the scattering geometry itself and the field in the local frame as discussed in the next section. See also further discussion by \citet{Judge07} and \citet{Schad18}.
Even without knowledge of the plasma density and temperature, however, some of the degenerate solutions may be ruled out for particular scattering angles based on the relationship between the sign of $\sigma^2_0$ and $\theta_B$, given by the ($3 \cos^2{\theta_B} - 1$) term discussed above \citep{Judge07}. At $\theta_{B} \approx 54.74^{\circ}$ (and ${\approx}125.26^{\circ}$), referred to as the Van Vleck angle, the atomic alignment changes sign. At each scattering angle along the line of sight, the solution pairs ($\sigma^2_0$, $\theta_B$) may be valid or determined to be invalid based on the sign relationship, or as discussed in \citet{Schad18}, ruled out if the scattering angle is unphysically large given the observations.
\begin{figure}
\centering
\includegraphics[width=3.3in]{all_solutions_3.pdf}
\caption{Plot of four degenerate solutions shown in the LOS frame of reference. Scattering angle is $\theta_{obs} = 90^{\circ}$ so the emission is located in the plane perpendicular to the LOS and bisecting the Sun.
The input field is (B, $\Theta_B$, $\gamma_B$) = (25 G, 35\deg, -59\deg) and corresponds one of the arrows shown in blue. The other blue arrow corresponds to the 180\deg ambiguous solution due to the lack of orientation in the reference direction for the linear polarization (r.d.l.p). The red pair of arrows represents the degenerate solutions due to the Van Vleck ambiguity. The dashed lines represent the projections of the solutions on the plane perpendicular to the line of sight.}
\label{fig:all_solutions}
\end{figure}
\subsection{Combining multi-line M1 inversions with other plasma diagnostics}
Combining multi-line M1 inversions with spectroscopic plasma diagnostics for density and temperature provides an opportunity to further constrain the ambiguity of the M1 inversion, as discussed by \cite{Judge13}, as well as a means to cross-validate numerical calculations of the atomic alignment with model independent empirical determinations. In addition, it is anticipated a simultaneous model fit of two or more M1 lines (as long as at least one of the lines has $F\ne0$) will further improve resilience of the solution to observational noise. Once again, the atomic alignment depends on electron density ($n_e$) and temperature ($T_e$), height above the photosphere and the magnetic field orientation relative to the local radial direction $\theta_B$ (see Figure \ref{fig:sigma_density}). Emission line ratios of both ultraviolet and infrared transitions may be used as plasma diagnostics to independently measure $n_e$ and $T_e$ \citep[e.g. see reviews by][]{DelZanna18a,delZanna18b}. This information can be used, in addition to knowledge of the projected height above the limb, to determine the scattering angle consistent with the multi-line M1 inverted parameters. Separately, the atomic alignment can be numerically calculated given the inferred plasma parameters, geometry, and field direction and therefore compared with the inferred value of $\sigma^2_0$ as a way to benchmark the theoretical calculations. However, we note that very high signal-to-noise is required to precisely determine the alignment (see Figure~\ref{fig:negative_sol}).
\begin{figure}
\centering
\includegraphics[width=3.3in]{sigma_density2.pdf}
\caption{Variation of the atomic alignment for the \ion{Fe}{13} 10747 \AA{} line with density, temperature and magnetic field inclination $\theta_B$ with respect to the incident radiation axis of symmetry. The atomic alignment is calculated at a height of 0.1 \Rsun (thin lines) and 0.3 \Rsun (thick lines).}
\label{fig:sigma_density}
\end{figure}
\subsection{A single line measurement of \texorpdfstring{$B_{LOS}$ at a single point}{B LOS at a single point}}\label{blos}
While so far we have focused on multi-line diagnostics, the formalism we presented in Section \ref{analytic_solution} also helps identify diagnostic techniques available for single M1 line spectropolarimetric observations. In particular, it has been known since \cite{Casini99} that the atomic alignment plays a role in the circular polarized amplitude and that the typical weak field magnetograph formula cannot be directly applied to measure the longitudinal component of magnetic field (\textit{i.e.} along the line-of-sight (LOS)) without an associated scale error. Interestingly, \cite{Plowman14} demonstrated via his Equation 14 that the scale error could by mitigated by using the observed linear polarized amplitude as a correction. In our generalization of this approach, the updated form of this relation is given via Equation \ref{eqB} as
\begin{align}
B^{obs}_{LOS} &= \frac{h}{\mu_B}\frac{-V}{\left[(I \pm L)\bar{g} \pm \frac{2}{3}F\frac{L}{\sin^{2}{\Theta_{B}}}\right]}. \label{eqBLOS}
\end{align}
where $B^{obs}_{LOS}$ is the observationally determined longitudinal magnetic field strength.
Two solutions for $B^{obs}_{LOS}$ are obtained which correspond to the positive and negative branches of equation \ref{eqBLOS}. For lines with $F=0$ (left panels in Figure \ref{fig:blos_measurement}), the solution branch that matches the sign of the atomic alignment accurately measures the input magnetic field ($B^{in}_{LOS}$) independent of $\Theta^{in}_B$ and $\sigma^{2in}_0$. The magnetic field measured for the degenerate solution over-predicts the input magnetic field by an amount that increases with both $\Theta^{in}_B$ and $\sigma^{2in}_0$ so fields more tangential to the LOS have a larger bias than fields parallel to the LOS.
An example of this effect can be observed for the example discussed in Section \ref{sec:noise}.
Note how for the negative solution branch (Figure~\ref{fig:negative_sol}) the inverted value of $B_{LOS}$ correlates much more tightly with the input value than the positive solution branch (Figure~\ref{fig:positive_sol}). As the F value for \ion{Fe}{13} 10747 \AA{} is zero, V and L measurements provide tight constraints on $B_{LOS}$ via Equation~\ref{eqBLOS} that are independent of $\Theta_{B}$ and track the true value of $B_{LOS}$ when the chosen solution matches the sign of the input alignment, in this case the negative branch. For this example the bias is small since the atomic alignment ($\sigma^2_{01}=-0.021$) is small compared to the alignment values used to produce Figure \ref{fig:blos_measurement}.
Meanwhile, for transitions with $F\ne0$ (right panels in Figure \ref{fig:blos_measurement}), both solutions for $B^{obs}_{LOS}$ are functions of $\Theta^{in}_B$ and $\sigma^{2in}_0$. Therefore, in principle, without a priori knowledge of these parameters both the measured values for $B^{obs}_{LOS}$ will have a bias relative to the true value $B^{in}_{LOS}$. It is possible to show that the bias is minimized at all values of $\Theta^{in}_B$ if a guess value of $\Theta^g_B=90^{\circ}$ is used in equation \ref{eqBLOS}. Other guess values for $\Theta^{g}_B$ will improve agreement locally but induce larger errors at all other angles (e.g. see dashed lines with $\Theta^g_B=45^{\circ}$ in the right panels of Figure \ref{fig:blos_measurement}). As discussed by \citet{Casini99} the weak field approximation or magnetograph formula (MF) relating $B_{LOS}$ to the Stokes measurements:
\begin{align}
B^{obs}_{LOS,MF} &= \frac{h}{\mu_B\bar{g}}\left(\frac{-V}{I}\right) \label{eqMF}
\end{align}
biases the measurements by an amount that depends on the atomic alignment since it effectively assumes $L=0$. Plots for the value of $B^{obs}_{LOS}$ using equation \ref{eqMF} are shown in blue in Figure \ref{fig:blos_measurement} and the values fall between the two solutions.
\begin{figure}
\centering
\includegraphics[width=3.3in]{blos_measurement_6.pdf}
\caption{Angle dependent ratios of the measured (or ``observed'') line of sight magnetic field $B^{obs}_{LOS}$ to the true input magnetic field $B^{in}_{LOS}$. The panels on the left correspond to measurements for a line with $F=0$ while the panels on the right correspond to a line with $F=0.5$, with the top (bottom) panels having positive (negative) input atomic alignments $\sigma^{2in}_0$. The black (green) curves represent $B^{obs}_{LOS}$ values calculated with the positive (negative) sign in equation \ref{eqBLOS} and having set $\Theta^g_B=90^{\circ}$ which minimizes the bias at all values for $\Theta_B$. For comparison the dashed lines assume $\Theta^g_B=45^{\circ}$. The blue curves represent $B^{obs}_{LOS}$ calculated using the magnetograph formula (MF, equation \ref{eqMF}).}
\label{fig:blos_measurement}
\end{figure}
\subsection{Approaches for implementation and limits to the application of the single point approximation}
\label{sec:implementation}
A reliable implementation of the single point inversion, using for example observations to be made available by the upcoming DKIST instruments, relies not only on using appropriate line combinations and sufficient single-to-noise but also on techniques to target and isolate regions of emission along the line of sight, which may be aided by observational checks on the validity of the single point approximation to a given observation.
Coronal loop structures provide one viable target for applying the single point approximation as has been done by many authors who study the plasma properties along loops using spectroscopic and spectro-imaging techniques \citep[e.g.][]{Aschwanden99, Brooks12, Xie17}. In these studies the importance of background/foreground emission subtraction is well known \citep[see][and references within]{Terzo10}. The intensity contrast of loops observed by TRACE (using the most sensitive filter for each loop) was generally found to be $\sim15-20\%$ \citep{Aschwanden05}. Warm ($\sim$1-2 MK) active region loops tend to have higher densities compared to diffuse emission surrounding the loops \citep{Brooks12, Brooks19}. As seen in Figure \ref{fig:sigma_density}, the atomic alignment decreases sharply with increases in the density (the temperature dependence is much smaller). Therefore, although loops have increased density and correspondingly larger brightness, the linearly polarized signals may be lower than for the background plasma. As such, the single-to-noise requirements become more demanding in order to implement a reliable background subtraction technique, such as fitting the cross section of the polarized emission profile across the loop. To understand how best to achieve this subtraction, synthesizing polarized emission through high resolution magnetohydrodynamic (MHD) simulations of the active corona \citep[e.g.][]{Rempel17} and then testing methods may be helpful.
Given a particular observation, one test for the validity of the single point approximation is provided by the two independent measurements of projected linear polarization angle. Since M1 lines are in the saturated Hanle regime, the angles $\gamma_B$ measured with different lines emitted from the same region should be equal. The condition is necessary but not sufficient because the possibility exists that background linear polarization may satisfy this condition while obscuring the signal. Resulting signals that have background subtraction applied should also satisfy this condition.
\section{Summary and outlook}
In this article, we have presented a generalization of the dual-line inversion algorithm for the vector magnetic field, proposed by \citet{Plowman14} for the \ion{Fe}{13} 10747, 10798 \AA{} line pair, to other combinations of magnetic dipole (M1) lines. A number of results that emerged from the analysis:
\begin{enumerate}
\item Assuming emission is concentrated at a single point in the corona it is possible to combine polarized observations from two or more M1 lines to invert for the vector magnetic field. However, only some combinations of M1 lines contain non-degenerate information such that a finite set of degenerate solutions can be calculated. Transitions between levels with equal upper/lower level Land\'e factors or $J=1 \rightarrow 0$ transitions cannot be used without adding another line where these conditions do not hold.
\item One pair of lines that satisfies this requirement is \ion{Fe}{13} 10747 \AA{} and \ion{Si}{10} 14301 \AA{}. Using atomic modelling of the polarized emission coefficients under realistic physical conditions for a coronal loop, we showed that the inclination of the magnetic field with respect to the line of sight, $\Theta_B$, and the magnetic field strength B can be recovered for photon noise levels $\sim10^{-4}I_{peak}$ if the magnetic field is oriented close to the line of sight.
\item In general the method produces four degenerate magnetic field solutions in the line of sight frame of reference consistent with the Van Vleck and 180\deg{} ambiguities. However, the number of allowed solutions may be smaller depending on the field geometry and the emission lines selected. Independent measurements of the height of emission (through stereoscopy) can be used together with an atomic model and the measured atomic alignment to determine the density and temperature of the emission region. Conversely, measurements of density and temperature (through coordinated observations with diagnostic UV and IR lines) can be used to constrain the height of emission.
\item The line of sight magnetic field $B_{LOS}$, obtained from polarized measurements of a single M1 line, is robust to noise but can be measured without bias only for M1 lines that have equal Land\'e factors for the upper and lower levels, or $J=1 \rightarrow 0$ transitions. Other M1 lines that do not satisfy these conditions will give a biased measurement value for $B_{LOS}$ that varies based on the inclination of the magnetic field with respect to the line of sight $\Theta_B$. However, this bias can be minimized for all values of $\Theta_B$ to better than 10\% even for large values of the atomic alignment if the inclination is assumed to be $\Theta_B=90^{\circ}$.
\end{enumerate}
The coronagraphic polarimetric capabilities of the upcoming National Science Foundation's \textit{Daniel K. Inouye Solar Telescope} provides an unprecedented opportunity to observe M1 emission from the solar corona at high resolution and high sensitivity such that multi-line techniques for coronal magnetic field diagnostics, including the techniques developed here, can be put into practice.
\section{Acknowledgments}
We extend our thanks to Roberto Casini, Phil Judge, and Joe Plowman for providing feedback and comments on this work. The National Solar Observatory (NSO) is operated by the Association of Universities for Research in Astronomy, Inc. (AURA), under cooperative agreement with the National Science Foundation.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,107 |
#ifndef __LagrangianDSTest__
#define __LagrangianDSTest__
#include <cppunit/extensions/HelperMacros.h>
#include "LagrangianDS.hpp"
#include "RuntimeException.hpp"
class LagrangianDSTest : public CppUnit::TestFixture
{
private:
/** serialization hooks
*/
ACCEPT_SERIALIZATION(LagrangianDSTest);
// Name of the tests suite
CPPUNIT_TEST_SUITE(LagrangianDSTest);
// tests to be done ...
CPPUNIT_TEST(testBuildLagrangianDS1);
CPPUNIT_TEST(testBuildLagrangianDS4);
CPPUNIT_TEST(testBuildLagrangianDS5);
CPPUNIT_TEST_SUITE_END();
// \todo exception test
void testBuildLagrangianDS1();
void testBuildLagrangianDS4();
void testBuildLagrangianDS5();
//void testcomputeDS();
// Members
SP::SiconosVector q0, velocity0, u0;
SP::SiconosMatrix mass;
public:
void setUp();
void tearDown();
};
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,090 |
Q: Make a small UIImageView easy to be tapped I have a UITableView with rows.
Each row has a small UIImageView aligned to the right (a "bookmark" icon)
The UIimageView has a UITapGestureRecognizer associated.
cell.favoritedImageView.userInteractionEnabled = true
cell.favoritedImageView.addGestureRecognizer(gestureRecognizer)
The problem is that to actually tap it with the finger (in a real device), you have to use the tip of the finger and be very accurate, because the image is small.
If you miss tapping the imageView, the cell is tapped (didSelectRowAtIndexPath) and you end up executing a show-segue to another view, so you have to go back and try again (not cool)
Question: what is the best way to solve this? I want it to be easy to be tapped.
I have some ideas:
*
*Create a larger image with transparent surrounding (ie: crop out with transparent background) -- downside is that I also use this image in other views, in which is not tappable, so I'd have to create two versions of the image
*Put the image inside a UIView and make the UIView big and tappable instead of the UIImageView
*Add padding to the UIImageView (will this work? or the padding is not recognized in the UITapGestureRecognizer?)
A: Per your own suggestion, you should create a transparent view that is much larger and attach the UITapGestureRecognizer to the view and then nest your smaller image within the view. That way appearances are the same, but you handle a much larger area for the tap to be recognized with selecting the cell.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 265 |
Millennia later, it had become an important coast-to-coast throughway. In the 18th century a military road linked the forts of Fort William, Fort Augustus, and Inverness. Later the Caledonian Canal provided a passage through the glen linking the west coast to the east coast. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,448 |
The Antillian snake eel (Ophichthus spinicauda, also known commonly as the spinefin snake eel or the banded snake eel) is an eel in the family Ophichthidae (worm/snake eels). It was described by John Roxborough Norman in 1922. It is a marine, deep water-dwelling eel which is known from the western central Atlantic Ocean, including Cuba, Puerto Rico, Trinidad-Tobago, and Venezuela. It is known to dwell at a maximum depth of 300 meters, and inhabits coastal waters. Males can reach a maximum total length of 107 centimeters, but more commonly reach a TL of 70 cm.
The Antillian snake eel is of minor commercial interest to fisheries.
References
Taxa named by John Roxborough Norman
Fish described in 1922
Ophichthus | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 789 |
Nigerians Demand Gun Rights After Boko Haram Jihad Spree
AFP PHOTO / HO / BOKO HARAM
The Nigerian military under President Muhammadu Buhari once again claimed to have defeated Boko Haram this week, only to have the group resurface and continue to carry out deadly attacks amid the Muslim holy month of Ramadan.
Buhari has been periodically claiming victory over the group since 2015.
In the days preceding the allegation that the military defeated Nigeria, reports surfaced highlighting the group's deadly atrocities and ongoing menace, including against children.
Echoing a victory lap over the Borno state-based Boko Haram in February, the Buhari administration maintains the military is now mainly focused on combating the international criminal gang known as Islamic State of West Africa Province (ISWAP), which allegedly broke away from Boko Haram in 2016 over leadership but consists of many of the same individuals.
Referring to Tukur Buratai, the chief of staff of the Nigerian Army, Premium Times reported on Thursday:
He explained that what is currently playing out in the North-east is the metamorphosis of ISWAP which is an attempt by a group of international criminal organizations to explore the loopholes created by the breakdown of law and order in some neighboring countries to perpetrate criminality in the West African sub-region.
[He] declared that just as Boko Haram was pursued out of the North-east, the current band of international criminals gangs operating under the guise of ISWAP will also be chased and hunted down.
"I never give up. Never retreat and never surrender is one of the secrets of my success. You must work hard, and hard work eventually pays.
Citing the International Crisis Organization, a widely cited non-governmental organization that monitors violent conflicts, the United Nations revealed on May 18 that '"an estimated 3,500-5,000" ISWAP jihadis "overshadow" Boko Haram, "which has roughly 1,500-2,000, and appears to have gained the military upper hand over the latter."
Unlike Boko Haram, the Islamic State branch has "made inroads among Muslim civilians by treating them better than its parent organization and by filling gaps in governance and service delivery."
Like its parent group, the Islamic State in Iraq and Syria, ISWAP jihadis, who are primarily targeting Nigerian troops, are employing their growing power and influence to establish "a proto-state in northern Nigeria."
Referring to the ISIS-West Africa branch, the Premium Times noted:
It has also caused real pain to the Nigerian military, its primary target, overrunning dozens of army bases and killing hundreds of soldiers since August 2018. As its name suggests, ISWAP is affiliated with the faded Islamic State, or ISIS, caliphate in Iraq and Syria, whose remnants count ISWAP victories as their own.
ISWAP appears to be working hard to gain greater favor from its namesake organization, and it has obtained some support already, notably in the form of training, though it is not clear how significant a boost this will afford.
Since launching its insurgency in 2009, Boko Haram, mainly based in the northeastern Nigerian state of Borno state, also tried to establish a sharia-compliant Islamic emirate in Nigeria, to no avail.
According to several independent and U.S. government assessments, many of the jihadis who fought alongside ISIS in Iraq and Syria fled to Africa and Afghanistan following the complete demise of their so-called caliphate at the hands of U.S.-backed local forces.
Thousands of ISIS jihadis remain in Iraq, Syria, and beyond.
The NGO's warnings about the rise of Islamic State-West Africa came as Nigerian lawmakers urged "Buhari to reverse the Executive Order revoking all gun certificates and licenses throughout the federation," Premium Times reported Thursday.
Opponents of the executive order argue that it "strips innocent people of the right to life and self-defense" against jihadi groups like Boko Haram and others, citing the rise of insecurity across the country where Muslim Fulani herders are also causing havoc.
Lawmaker Nnenna Ukeje, who is proposing the changes to the gun restrictions, noted "that there was an increase in the number of deaths resulting from terrorism, kidnapping and banditry; as well as other violent crimes in the country," the Premium Times said, adding:
Ms. Ukeje said that in response to the local and international outcry, the president signed an executive order to remove, revoke, and banish all gun certificates and licenses throughout the federation.
She said that the order was targeted at legal gun owners while there appeared to be no institutional actionable policy or plan to mop up small arms and light weapons in the country.
Ukeje acknowledged that most crimes are carried out with illegally obtained guns.
AfricaNational SecurityISILISISIslamic State-West Africa ProvinceISWAPMuhammadu BuhariNigeria | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,187 |
Wydarzenia
Europa
Ingólfur Arnarson, uznany przez średniowieczne dzieło Landnámabók za pierwszego stałego osadnika na Islandii, wylądował na wyspie i założył Reykjavik
pokój w Forchheim – Ludwik II Niemiecki uznał Świętopełka wielkomorawskiego
Serbia przyjęła chrześcijaństwo z Bizancjum.
Początek osadnictwa na Islandii przez Normanów
874 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,048 |
\section{Introduction}
The control of single-electron spin in nanoscale devices is a key element in the
field of spintronics, where the spin degree of freedom is used for information
transfer and processing. \
The vast majority of proposed spin devices are based on semiconductors
\cite{kane,friesen,friesen2,datadas}, where single-electron spin has been probed to
be controllable by means of voltages applied to electrostatic gates \cite{petta}.
A promising technology for the implementation of quantum computation (QC)
involves the storage of quantum information in the spin of electrons in
quantum dots (QDs). \ The key requirement is that the lifetime of the spins
is long compared with the time required for the operation of logic gates. \
This has motivated the development of QDs in Si-based materials \cite{nakul}, where spin-orbit
coupling (SOC) is weak and isotopic enrichment can eliminate hyperfine coupling (HC). \
In such scenario, processes limiting QC are dominated by
SOC resulting from spatial inversion asymmetry. \
In typical solid state systems, macroscopic electric fields cause structure inversion
asymmetry (SIA), giving rise to Rashba-type terms \cite{rashba} of the form:
\[
H_\mathrm{R}= \alpha (\hat \sigma_x \hat k_y -\hat \sigma_y \hat k_x),
\]
whereas fields resulting from the lack of an inversion center lead to
bulk inversion asymmetry (BIA) and to the Dresselhaus term,
\cite{dresselhaus}:
\[
H_\mathrm{D} = \beta (\hat \sigma_x \hat k_x -\hat \sigma_y \hat k_y).
\]
Here, $\sigma_i$ and $ k_i$ denote spin and momentum operators, respectivelly, and
$\alpha$, $\beta$ are the Rashba and Dresselhaus coupling constants, respectivelly. \
Although both SOC contributions have been noted for decades,
their absolute value
have been measured simultaneously only recently \cite{ensslin}. \
The different theoretical models show controversy \cite{hu},
with debated estimations for Rashba or Dresselhaus related parameters.
In this paper we calculate spin-flip rates for typical Si QD via higher-energy
virtual state, involving also phonon emission. \
We define the regime for which SOC is the dominant
source of relaxation as a function of $^{29}$Si isotope concentration. \
We utilize in our calculations the SOC parameters extracted numerically
using for first time a model that includes atomic crystal
symmetry, spin and interfaces built into the basis representation \cite{nextprl},
that aims to clarify the existing controversy.
This paper is organized as follows:
Sec. \ref{sec1} contains a description of the model:
we describe the electron-phonon interaction, and the spin-flip
mechanism, namely SOC and hyperfine coupling (HC).
In Sec. \ref{sec2} we present our results,
and finally, Sec. \ref{sec3} is devoted to conclusions.
\section{Methods}
\label{sec1}
\subsection{Model}
We consider a QD formed in a two dimensional (2D) heterostructure
with parabolic confinement. \
Spin-flip is considered via an
orbital (or valley) state with energy
$\hbar\omega_0$, as depicted in Fig. \ref{figscheme}. \
\begin{figure}[!hbt]
\centering\includegraphics[angle=0, width = .2\textwidth]{./figs/level_scheme}
\caption{\footnotesize
Schematic representation of a spin-flip process for a Si QD with one electron.
}
\label{figscheme}
\end{figure}
In order to calculate the relevant transition rates for spin relaxation,
we consider a perturbation given in general by
$\delta H = H_{\mathrm {sf}} + H_{\mathrm{ph}}$,
$H_{\mathrm{ph}}$ corresponding to the electron-phonon
coupling, and $H_{\mathrm {sf}}$ to the mechanism causing a spin
flip in the dot. \ The amplitude of a
spin-flip event in the ground state then is given in second order
perturbation theory,
\begin{equation}
\label{eq:st}
\langle 1_\uparrow | \delta H |1_\downarrow \rangle \thickapprox
\sum_n\left(
\frac{\langle 1_\uparrow | H_{\mathrm{ph}} |n_\uparrow \rangle \langle n_\uparrow| H_{\mathrm{sf}} |1_\downarrow\rangle}
{E_n - E_1 + \Delta }
\right),
\end{equation}
where $\Delta$ is the energy exchanged with the bath. \ Here,
$\uparrow,\downarrow$ denote the spin state and $n$ is the orbital degree of freedom. \
The orbitals are described in terms of Fock-Darwin states. \
The ground-state spin-flip rate $|1_\uparrow\rangle$ to $|1_\downarrow\rangle$ is then given by Fermi's
golden rule:
\begin{equation}
\label{eq:st2}
\Gamma_{\uparrow,\downarrow} =
\frac{2\pi}{\hbar}|\langle 1\uparrow | \delta H |1\downarrow\rangle|^2 \delta(E_i-E_f).
\end{equation}
In this notation, $|1\downarrow\rangle$ denotes the initial state of electron, nuclei
and phonons, $|1\downarrow\rangle \to |i\rangle$ =
$|1\downarrow\rangle \otimes |i_n\rangle \otimes |i_{\mathrm{ph}}\rangle$, likewise
$|1 \uparrow\rangle \to |f\rangle$ =
$|1 \uparrow\rangle\otimes|f_n\rangle \otimes |f_{\mathrm{ph}}\rangle$. \
Inserting Eq. (\ref{eq:st2}) into (\ref{eq:st}) we obtain (setting $\Delta\to$0):
\begin{equation}
\label{eq:st3}
\Gamma_{\uparrow,\downarrow} =
\frac{2\pi}{\hbar}
|\langle \langle f_{\mathrm{ph}} | H_{\mathrm{ph}}
|i_{\mathrm{ph}}\rangle|^2 \delta(E_i-E_f)
\frac{|\langle 1\uparrow|H_{\mathrm{sf}} |1\downarrow \rangle|^2}
{(\hbar\omega_0)^2}
\end{equation}
We note that the phonon and the spin-flip parts can be factorized, so we
define $\Gamma_{\uparrow,\downarrow} = \Gamma_{\mathrm{ph}}\times \zeta^2$, with
\begin{equation}
\Gamma_{\mathrm{ph}} = \frac{2\pi}{\hbar}|\langle i | H_{\mathrm{ph}} |f \rangle|^2 \delta(E_i-E_f); \quad
\zeta^2 = \frac{|\langle 1\uparrow|H_{\mathrm{sf}} |1 \downarrow\rangle|^2}{(\hbar\omega_0)^2
\label{eq:gammaphonon1}
\end{equation}
Hence, $\Gamma_{\mathrm{ph}}$ and $\zeta$ can be treated
separately. \ $\Gamma_{\mathrm{ph}}$ describes the phonon-emission
process, coupling two orbitals via electron-phonon deformation potential,
and $\zeta$ includes the spin-flip process. We give next a detailed description
of both terms. \\
\subsection{Phonon-electron coupling}
For Si under compressive stress along [001], the electron interacts
with a phonon of momentum $\bm{q}$ via deformation potentials \cite{charlie,
herring, hasegawa}. The Hamiltonian reads:
\begin{eqnarray}
\label{eq:Heph1}
H_{\mathrm{ph}} &=& \sum_s\sum_{\bm q}
i[a^*_{qs}e^{-i{\bm q r}} + a_{qs}e^{i{\bm q r}} ]
q(\Xi_d \hat e_x^s\hat q_x + \Xi_d \hat e_y^s\hat q_y + \nonumber\\&&
(\Xi_d + \Xi_u)\hat e_z^s\hat q_z)
\end{eqnarray}
where $\langle n_q-1| a_q | n_q \rangle = \sqrt{(\hbar n_q/2M_c\omega_q)}$,
$M_c$ is the mass of the unit cell and $\hbar\omega_q$ the phonon energy. \
Here, $s$ denotes the polarization of the phonon (two transverse and one
longitudinal), $\bm q$ is the wavevector, $\Xi_u$ and $\Xi_d$ are the
electron-phonon coupling parameters in Si (see table \ref{table1}). \ This
is slightly simpler than the corresponding Hamiltonian in GaAs because of
the absence of the piezoelectric coupling in Si. \
In order to calculate $\Gamma_{\mathrm{ph}}$, we use the
electric dipole (ED) approximation, $e^{i{\bm q r}} \thickapprox 1 + i {\bm q r}$ (valid in
the range of ~0.1-0.2 meV, which is roughly the phonon energy involved in
this relaxation process). \\
For the process depicted in Fig. \ref{figscheme}, mixing
of $s$-orbitals occur at 0-th order in ED (to lowest order). \ We summarize in table
\ref{table1} the parameters relevant for the computation of $\Gamma_{\uparrow,\downarrow}$
used throughout the text.
\begin{table}[!hbt]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$\Xi_u$[eV]& $\Xi_d$[eV] & $\rho_{\mathrm {Si}}[$kg$/$m$^{3}$]& $v_l$[m/s] & $v_t$[m/s]
&$A$ [$\mu$eV$\cdot$nm$^{3}$] & $\langle\rho^2\rangle$[nm$^2$]
\\
\hline
9.29 &-10.7 & 2330 &9330&5420
& 0.2&4$\times$10$^2
\\
\hline
\end{tabular}
\caption{\footnotesize Physical constants and material parameters for Si
and the QD as in \cite{nakul}.}
\label{table1}
\end{table}
Using the 0-th order ED approximation,
$e^{i{\bm q r}} \thickapprox 1 $, we obtain
for the electron-phonon coupling:
\[
\widehat H_{\mathrm{ph}} \thickapprox
\sum_q(a_q + a_q^\dag)|q|
[\Xi_d\hat e_xq_x +\Xi_d\hat e_yq_y +(\Xi_d + \Xi_u)\hat e_zq_z],
\]
which, using (\ref{eq:gammaphonon1}), gives the rate:
\begin{eqnarray}
\label{eq:matrixeldir}
\Gamma_{\mathrm{ph}}^{\mathrm{D}}
&\thickapprox &
\frac{(n_q + 1)}{2\rho_{\mathrm {Si}}(2\pi)^2}
\sum_s\int_0^{2\pi}\mathrm{d}\varphi \int_0^\pi \mathrm{d}(\cos{\theta})\int_0^\infty \mathrm{d} q_s
\frac{q_s^4}{\omega_{q_s}}
[\Xi_d\hat e_x^s\hat q_x + \nonumber\\
&& \Xi_d\hat e_y^s\hat q_y
+(\Xi_d + \Xi_u)\hat e_z^s\hat q_z]^2 \delta(\hbar\omega_{q_s}-\hbar\omega_0).
\end{eqnarray}
Here, $n_q$+1 $\thickapprox$ 1 in
the range of low temperatures considered, and $\rho_{Si}$
is the density of Si (see table \ref{table1}). \
$\hat e_i$ is the $i$-th component of the unitary polarization vectors
and $s$ denotes the polarization,
$s$=l, t$_1$, t$_2$:
\begin{eqnarray}
\label{eq:polarvecs}
\hat e^{l} & = & \sin{\theta} \cos{\varphi}\hat u_x +
\sin{\varphi} \hat u_y -\cos{\theta} \cos{\varphi}\hat u_z \nonumber \\
\hat e^{t_1} & = & \sin{\theta} \sin{\varphi}\hat u_x -
\cos{\varphi} \hat u_y -\cos{\theta} \sin{\varphi}\hat u_z \nonumber \\
\hat e^{t_2} & = & \cos{\theta} \hat u_x + \sin{\theta} \hat u_z.
\end{eqnarray}
and $\hat q = (\sin{\theta} \cos{\varphi}, \sin{\theta} \sin{\varphi},\cos{\theta})$.
To integrate Eq. (\ref{eq:matrixeldir}), we assume an isotropic phonon
spectrum, $E_{\mathrm{ph}}= \hbar \omega_{qs}$ with a dispersion relation
$\omega_{qs} = v_s q$, $v_s$ being the sound velocity of the mode $s$. This
gives:
\[
\int_0^\infty \frac{dq q_s^4}{\omega_{q_s}}
\delta(\hbar\omega_{q_s}-\hbar\omega_0) =
\frac{(\hbar\omega_0)^3}{\hbar^4 v_s^5}.
\]
Using the above result, the polarization vectors
of (\ref{eq:polarvecs}), rearranging (\ref{eq:matrixeldir}), and performing
the angular integral, we obtain a compact expression for the rate:
\begin{eqnarray}
\label{eq:gammaphD}
\Gamma_{\mathrm{ph}}^{\mathrm{D}}
&\thickapprox&
\left(\frac{\hbar\omega_0}{\hbar}\right)^3 \frac{1}{2\pi\hbar \rho_{Si}}\frac{1}{15}
\left[
\frac{15\Xi_d^2 + 10\Xi_d\Xi_u + 3\Xi_u^2}{v_l^5} +\right.\nonumber\\ &&\left.
\frac{3\Xi_d^2 + 4\Xi_d\Xi_u + 2\Xi_u^2}{v_t^5}
\right].
\end{eqnarray}
\subsection{Spin orbit coupling}
As pointed out before, spin-flips can be provided by two different mechanisms,
the HC with the $^{29}$Si nuclei and the SO coupling. \
For an isotopically-purified sample, we expect SO to be the dominant
spin-flip mechanism. \ Hence, we first evaluate SO mediated spin-flip
and then consider the $^{29}$Si concentration at which the HC mediated
spin-flip rate becomes comparable to the SO mediated one. \ \
We express the Hamiltonian in the convenient phase coordinates ($q_1$,$q_2$,$p_1$,$p_2$)
\cite{galkin}:
\begin{equation}
\label{eq:hosc2}
\hat H_{0} = \frac{1}{2m}
\left(
\hat p_1 ^2 + \hat p_2^2\right) +
\frac{m^*}{2}(\omega_1^2\hat q_1^2 + \omega_2^2\hat q_2^2) +
\frac{1}{2}g\mu_BB\hat\sigma_z,
\end{equation}
with $\omega_{1,2}=(\omega_0^2+\omega_c^2)^{1/2} \pm \omega_c$ and $\omega_c = eB/m $. \
\begin{figure}[!hbt]
\centering\includegraphics[angle=0, width = .35\textwidth]{./figs/rotated}
\caption{\footnotesize
(a) Schematic representation of a spin-flip process for a Si QD with one electron.
(b) Spin direction for the eigenfunctions of Rashba and Dresselhaus contributions:
for $\bm{p}\parallel[110], [1\bar10]$, the effective magnetic fields are parallel.
}
\label{figrot}
\end{figure}
We add as a perturbation the Rashba and
the Dresselhaus SOC terms in a convenient basis rotated by 45$^\circ$
(see Fig. \ref{figrot}):
\begin{equation}
\label{eq:soc_r_d}
\hat H_{\mathrm{SO}} =\frac{1}{\hbar}\left[
(\beta + \alpha)\hat\sigma_+ \hat p_- +
(\beta -\alpha) \hat \sigma_-\hat p_+
\right].
\end{equation}
We note that the SOC term does not commute with $\widehat H_0$. \
This implies that the spin does not remain fixed during the motion of the electron,
apart for an apparent homogeneous shift of the momentum to first order, which would
not change observables (other than just a shift in the energy levels). \
To get rid of such terms fixed by gauge invariance,
it is convenient to apply the transformation of Aleiner {\it{et al.}}, \cite{aleiner}
$\widehat U = \mathrm{exp}\{i\gamma_1\hat q_+\hat \sigma_- + i \gamma_2\hat q_-\hat \sigma_+ \}$,
($\hat q_\pm$ is the position operator along the directions $\hat e_\pm$),
which allows to see how the levels split in the presence of $\widehat H_{SO}$. \
Using the Zassenhaus formula,
$e^{\hat A + \hat B} \simeq e^{\hat A } e^{\hat B} e^{-\frac{1}{2}[\hat A,\hat B]}\cdots$
we factorize $U$ to second order in SOC:
\begin{equation}
\label{eq:U1}
\widehat U = e^{i(\gamma_1\hat q_+\hat \sigma_- + \gamma_2\hat q_-\hat \sigma_+ ) } \simeq
e^{i\gamma_1\hat q_+\hat \sigma_- } e^{i \gamma_2\hat q_-\hat \sigma_+ }
e^{-\frac{1}{2}\gamma_1\gamma_2\hat q_+ \hat q_-[\hat\sigma_-,\hat \sigma_+] }.
\end{equation}
Retaining terms to third order in SOC, the momentum transform as:
\[
\widehat U^\dag \widehat P_+ \widehat U \simeq
\widehat P_+ + \hbar\gamma_1\hat \sigma_- -\hbar\gamma_1\gamma_2\hat q_- \hat\sigma_z -
2\hbar\gamma_1^2\gamma_2\hat q_-\hat q_+ \hat\sigma_+,
\]
and likewise:
\[
\widehat U^\dag \widehat P_- \widehat U \simeq
\widehat P_- + \hbar\gamma_2\hat \sigma_+ +\hbar\gamma_1\gamma_2\hat q_+ \hat\sigma_z -
2\hbar\gamma_1\gamma_2^2\hat q_-\hat q_+ \hat\sigma_-.
\]
so that the total Hamiltonian, $\widehat H = \widehat H_0 + \widehat H_{\mathrm{SO}}$, transforms as
$\widehat H^\prime =
\widehat U^\dag \widehat H \widehat U$. \ Considering terms up to third-order in SOC:
\begin{eqnarray}
\label{eq:htransformed2}
\widehat H^\prime &\simeq&
\widehat H_0+
\frac{\hbar}{m l_1 l_2}[\hat q_+ \hat p_- - \hat q_- \hat p_+ ]\hat \sigma_z +
\nonumber\\&&
\frac{2\hbar}{m l_1 l_2}\hat q_+ \hat q_-
\left[
\frac{\hat p_+\hat \sigma_+}{l_1} - \frac{\hat p_-\hat \sigma_-}{l_2}
\right]= \widehat H_0+\widehat H_1+\widehat H_2,
\end{eqnarray}
where we have defined $\gamma_1$=${m(\alpha -\beta)}/{\hbar} \equiv l_1^{-1}$ and
$\gamma_2$ = ${-m(\alpha +\beta)}/{\hbar} \equiv -l_2^{-1},$ with
$l_{1,2}$ characterizing the length scale associated with the strength of
the SOC for electrons moving along the crystallographic directions. \
This choice allows us to cancel out the linear terms in momentum. \
The second term of Eq. (\ref{eq:htransformed2}) gives an effective magnetic
field that can be
expressed as:
\begin{equation}
\label{eq:h1}
\widehat H_1 = i h_
[\hat a_1\hat a_y^\dag - \hat a_y \hat a_x^\dag]\hat\sigma_z, \quad
\end{equation}
where we have defined:
\[
h_1 = 2m^*(\alpha^2-\beta^2)\left(\sqrt{\frac{\omega_2}{\omega_1}}
+\sqrt{\frac{\omega_1}{\omega_2}}\right) .
\]
We note that $\widehat H_1 $ does not break the Kramers degeneracy,
and hence, does not contribute to spin-flip. \ However, the last term of
Eq. (\ref{eq:htransformed2}) does break Kramers degeneracy, allowing spin-flip
terms. \ In second quantization, we have:
\begin{eqnarray}
\label{eq:h22ndq}
\widehat H_2 &=&
i\gamma_y[\hat a_x^\dag\hat a_x^\dag\hat a_y +
\hat a_y^\dag (1-\hat n_x) - \hat a_y\hat n_x - \hat a_y^\dag \hat a_x\hat a_x ]\hat \sigma_x \nonumber\\
&&-i\gamma_y [\hat a_y^\dag\hat a_y^\dag\hat a_x +
\hat a_x^\dag (1-\hat n_y) - \hat a_x\hat n_y -\hat a_x^\dag\hat a_y\hat a_y ]\hat \sigma_y,
\end{eqnarray}
where we have defined:
\[
\gamma_y=(\beta-\alpha)^2(\beta+\alpha)m\sqrt{\frac{2m}{\hbar\omega_1}};
\
\gamma_x=(\beta+\alpha)^2(\beta-\alpha)m\sqrt{\frac{2m}{\hbar\omega_2}}.
\]
We then conclude that, for SOC, $\zeta_{\mathrm{SO}} \thickapprox \gamma_i^2/\hbar\omega$,
from which we can proceed to calculate $\Gamma_{\uparrow,\downarrow}$,
using (\ref{eq:h22ndq}) and (\ref{eq:gammaphD}) in (\ref{eq:gammaphonon1}). \
We emphasize that this term is third order in SOC, and applying Fermi
Golden's rule, the spin-flip rate will appear in sixth order. \ Hence,
$\Gamma_{\uparrow,\downarrow}$ is very sensitive to the SOC parameters,
so it is critical to determine them to a high degree of
precision.
\subsection{Hyperfine Coupling.}
For a electronic spin in a QD and in the presence of nuclear spins, a
contact interaction Hamiltonian can be assumed as a perturbation,
\begin{equation}
\label{eq:Hhf}
\widehat H_{\mathrm{hc}} = \sum_{i,j} \frac{4\mu_0}{3I}\mu_B\mu_I\eta
{\bm S}_i {\bm I}_j\delta(\bm r_i - \bm R_j)=
A\sum_{i,j}{\bm S}_i {\bm I}_j\delta(\bm r_i - \bm R_j)
\end{equation}
where ${\bm S}_i$ (${\bm I}_j$) and $\bm r_i$ ($\bm R_j$) denote the spin
and position of the $i^\mathrm{th}$ electron ($j^\mathrm{th}$ nuclei), and $A$ is the hyperfine
coupling constant. \ We now use (\ref{eq:Hhf}) as the spin-flip mechanism
to calculate $\zeta$ of Eq. (\ref{eq:gammaphonon1}):
\[
|\langle 1_\uparrow|H_{\mathrm{hc}} |2_\downarrow\rangle|^2 =
A^2\sum_j|\langle \uparrow|S^+I_j^-\delta(r-R_j)|\downarrow\rangle|^2.
\]
Next we substitute the sum over $j$ by an integral over the nuclei with
a density $C_n$:
\[
|\langle 1_\uparrow|H_{\mathrm{hc}} |2_\downarrow\rangle|^2 =
A^2C_n\int\mathrm{d}^3 R_j|\Phi_1(R_j)|^2 |\Phi_2(R_j)|^2
\]
which can be easily performed assuming Fock-Darwin states for $\Phi_i$,
resulting:
\begin{equation}
\label{eq:result1}
\zeta_{\mathrm{hc}}=\frac{3A^2C_n}{8V_{\mathrm{QD}}(\hbar\omega)^2 }
\end{equation}
Using (\ref{eq:result1}) together with Eq.
(\ref{eq:gammaphD}) in (\ref{eq:gammaphonon1}), we can thus obtain
the spin-flip rate due to HC.
Typical values for $\eta^{Si}$ of 186 have been reported \cite{shulman,abalmassov,paget},
yielding $A\thickapprox 2\times 10^{-7}$eV$\cdot $nm$^{3}$.
Only about 4$\%$ of the nuclei have spin, so $C_n \thickapprox
0.04\times 8/v_0 \thickapprox 2$nm$^{-3}$ ($v_0 \thickapprox 0.17$nm$^{3}$, is
the unit cell volume for Si).
It is important to note that $\Gamma _{\uparrow\downarrow}$ is proportional to $%
C_{n}, $ i.e., to the total number of nuclei $N_{n}$ with which the
electrons interact. \ This is consistent with the simple picture that the
relaxation rate is proportional to the mean-square fluctuations in the
random hyperfine field. \ Formulas for spin relaxation rates due to
HC that give an apparent proportionality to $N_{n}^{-1/2}$
are common in the literature, and have given rise to the incorrect notion
that some sort of motional narrowing is at work. \ This is not possible,
since the fluctuations in the nuclear spin system are slow. \ In any case
the rate must vanish as $N_{n}\rightarrow 0.$ \ These formulas are correct,
but they generally involve other parameters that actually vary with $N_{n}.$ \ \
\section{Results}
\label{sec2}
We considered a typical QD formed as in \cite{nakul}
with a level separation of $\hbar\omega_0$ = 0.2meV.
The $\alpha$ and $\beta$ parameters where extracted numerically using NEMO-3D
\cite{nemo} on nanoHUB.org computational resources \cite{nanohub}.
In NEMO-3D, atoms are represented explicitly in the {\it{sp$^3$d$^5$s$^*$} }
tight-binding model, and the valence force field (VFF) method is employed to
minimize strain \cite{strain}. NEMO-3D enables the calculation of localized
states on a QW and their in-plane dispersion relation with a very high degree
of precision, allowing to extract the splittings along the in-plane directions
in momentum space \cite{nextprl}. \
For now, we note that the $\alpha$ value is well defined and depends on
external electric fields applied to the sample, whereas the Dresselhaus, $\beta$,
depends strongly on the atomistic details of the interface. \
\begin{figure}[!hbt]
\centering\includegraphics[angle=0, width = .475\textwidth]{./figs/gammaofCnn_fig}
\caption{\footnotesize
$\Gamma_{\uparrow\downarrow}$ as a function of $^{29}$Si isotope
purification degree. \ The horizontal lines correspond to $\Gamma_{\uparrow\downarrow}$
due to SOC, for various values of Dresselhaus coupling parameter, $\beta$. \
}
\label{results1}
\end{figure}
Fig. \ref{results1} shows the calculated values for $\Gamma_{\uparrow\downarrow}$
via both mechanism, HC and SOC.
The parallel lines correspond to $\Gamma_{\uparrow\downarrow}^{\mathrm{SOC}}$ for
different possible values of $\beta$,
whereas the thick solid line represents the spin-flip due to HC mechanism,
$\Gamma_{\uparrow\downarrow}^{\mathrm{HC}}$, as a function of $^{29}$Si isotope
purification degree. \
Using our numerical value for $\alpha$ and a few different values for $\beta$,
we find that for natural Si, the HC dominates.
We note that these depend strongly on the sample, so a Si sample with
dominating SOC is possible. \
In particular, for a Si/SiGe heterostructure, the $\Gamma_{\uparrow\downarrow}^{\mathrm{HC}}$
value corresponds to the horizontal solid line of Fig. \ref{results1}. We can
see that a purification of about 80\% of the $^{29}$Si isotope will cause SOC
to become dominant. \ In contrast, for a similar QD formed in a Si membrane,
with purification of $\simeq$ 99\% we obtain SOC as the dominant mechanism,
increasing the relaxation time by almost two orders of magnitude.
\begin{figure}[!hbt]
\centering\includegraphics[angle=0, width = .44\textwidth]{./figs/purif_of_beta_fig}
\caption{\footnotesize Regions where SOC (shadded) or HC (light) are
the dominant mechanism of relaxation as a function of Dresselhaus parameter,
$\beta$, and degree of $^{29}$Si isotope purification:
for small $\beta$, a larger isotopic purification degree is needed, and hence,
larger relaxation times can be achieved.
}
\label{results2}
\end{figure}
Fig. \ref{results2} represents the regions for which SOC or HC are
the dominant spin-flip mechanism. \ It also summarizes the most relevant $\beta$
values found numerically: a typical QD defined in a Si/SiGe heterostructure
has $\beta\simeq$75m/s, for which HC is dominant up to a purification level of
81.5\%. A similar structure obtained on a Si membrane will show a larger Dresselhaus,
and hence, larger purification will be needed to have SOC as dominating mechanism.
\section{Conclusions}
\label{sec3}
In summary, we have calculated the dominant rates for phonon-assisted
spin-flip relaxation for the ground state of a single-electron
Si QD. \ $\Gamma_{\uparrow,\downarrow}$ is
found to be of the order of tens of seconds, very sensitive to SOC.
We observe that choosing a sample with small SOC, as pure Si membranes, can lead to
spin lifetimes of up to a few hours. \
QDs fabricated in Si/SiGe heterostructures are more limited by SOC,
and purifications of only about 80\% would give spin lifetimes of a few minutes.
\ Due to weak spin-orbit and HC, Si offers very long
coherence times, which are required for solid state qubits.
This work was supported by the Spanish
Ministry of Education and Science (MEC). We acknowledge support from
the Army Research Office (W911NF-04-1-0389) and the National Science Foundation (NSF-ECS-0524253)
and through the MRSEC-IRG1 at UW-Madison.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,574 |
Q: Editor and contributor roles not correct after adding function ** RESTATED **
I am creating a bridge. The work involves a default installation of WordPress and a plugin to bridge a forum and WordPress. The forum provides all the account information. The secondary usergroup of the forum is matched to the WP role. Now I am trying to retrieve the "role" selected in the options for that logged in user and add_role for the string. Inserting wp_die shows the correct role is being given to the logged in user. However, the wrong capabilities are being added. Everyone is getting admin.
// returns an array
var_dump( $XF->options['xf_user_role'] );
//returns string() "role such as administrator" (in quotes)
var_dump( $XF->options['xf_user_role'][$user_group] );
** UPDATED **:
Here is the latest partial code. There is no reason to really show filluserData
class XF_User_Data {
public static function fillUserData($visitor, $classObj, $user_id) {
// doing something and calls set_user_roles
}
public static function set_user_roles( &$classObj, $user_group_id, $secondary_group_ids ) {
/**
* Function set_user_roles(&$classObj, $user_group_id, $secondary_group_ids)
*
* This function sets the current user role based on the secondary group
*/
global $XF, $wp_roles;
$user_groups = explode( ',', $secondary_group_ids );
$user_groups[] = $user_group_id;
/**
* Loop through the user_roles and find the WP role equivalent for each
*/
$user_roles = array();
foreach( $user_groups as $user_group ) {
$user_roles[] = $XF->options['xf_user_role'][$user_group];
/**
* wp_die( 'This is the role:' . var_dump( $XF->options['xf_user_role'] ) );
*
* Returns:
*
* array(7) { [3]=> string(13) "administrator" [4]=> string(6) "author" [2]=> string(10) "subscriber" [1]=> string(10) "subscriber" [5]=> string(6) "author" [7]=> string(11) "contributor" [6]=> string(6) "editor" }
*/
/**
* wp_die( var_dump( $XF->options['xf_user_role'][$user_group] ) );
*
* Returns for user Ellie who has secondary group related to author
* string(6) "author"
*
* Returns Albert who has secondary group set to editor
* string(6) "editor"
*
* Returns Pascal who has secondary group related to contributor
* string(11) "contributor"
*
* Returns Gracie NO secondary group for registered members.
* NULL
*
*/
$new_roles = $XF->options['xf_user_role'][$user_group];
// wp_die( var_dump( $XF->options['xf_user_role'][$user_group] ) );
if( $new_roles == 'administrator' ) {
$capabilities = array (
'activate_plugins','delete_others_pages','delete_others_posts','delete_pages','delete_plugins','delete_posts','delete_private_pages','delete_private_posts','delete_published_pages','delete_published_posts','edit_dashboard','edit_files','edit_others_pages','edit_others_posts','edit_pages','edit_posts','edit_private_pages','edit_private_posts','edit_published_pages','edit_published_posts','edit_theme_options','export','import','list_users','manage_categories','manage_links','manage_options','moderate_comments','promote_users','publish_pages','publish_posts','read_private_pages','read_private_posts','read','remove_users','switch_themes','upload_files','create_product'
);
$user_id = $XF->visitor->get('user_id');
$user = new WP_User( $user_id );
foreach ( $capabilities as $cap ) {
$user->add_cap( $cap );
}
break;
}
elseif( $role == 'editor' ) {
$capabilities = array (
'delete_others_pages','delete_others_posts','delete_pages','delete_post','delete_private_pages','delete_private_posts','delete_published_pages','delete_published_posts','edit_others_pages','edit_others_posts','edit_pages','edit_posts', 'edit_private_pages','edit_private_posts','edit_published_pages','edit_published_posts','manage_categories','manage_links','moderate_comments','publish_pages','publish_posts','read','read_private_pages','read_private_posts'
);
$user_id = $XF->visitor->get('user_id');
$user = new WP_User( $user_id );
foreach ( $capabilities as $cap ) {
$user->add_cap( $cap );
}
break;
}
elseif( $new_roles == 'author' ) {
$capabilities = array(
'delete_posts', 'delete_published_posts','edit_posts', 'edit_published_posts','publish_posts', 'read', 'upload_files','upload_files'
);
$user_id = $XF->visitor->get('user_id');
$user = new WP_User( $user_id );
foreach ( $capabilities as $cap ) {
$user->add_cap( $cap );
}
break;
}
elseif( $new_roles == 'contributor' ) {
$capabilities = array( 'delete_posts', 'edit_posts', 'read' );
$user_id = $XF->visitor->get('user_id');
$user = new WP_User( $user_id );
foreach ( $capabilities as $cap ) {
$user->add_cap( $cap );
}
break;
}
elseif( $new_roles == '' || $new_roles == NULL || $new_roles == 'subscriber' ) {
$capabilities = array( 'read' );
$user_id = $XF->visitor->get('user_id');
$user = new WP_User( $user_id );
foreach ( $capabilities as $cap ) {
$user->add_cap( $cap );
}
break;
}
}
}
public static function create_new_wp_user( $new_role, $user_id ) {
if( null == username_exists( $user_id ) ) {
ini_set('memory_limit', '128M');
// Generate the password and create the user
$password = wp_generate_password( 12, false );
$user_id = wp_create_user( $email_address, $password, $email_address );
// Set the nickname
wp_update_user(
array(
'ID' => $user_id,
'nickname' => $email_address
)
);
// Set the role
$user = new WP_User( $user_id );
$user->set_role( $new_role );
// Email the user
wp_mail( $email_address, 'Welcome!', 'Your Password: ' . $password );
return;
} // end if
}
}
The wrong capabilities are being added to the user roles. For example, everyone is getting administrator capabilities.
What is wrong with this code?
PS. I am a neophyte and know just enough to be dangerous ;)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,600 |
'use strict';
function getSequenceTask(options, gulp, mode) {
var runSequence = require('run-sequence').use(gulp);
function sequenceTask(next) {
Object.assign(mode, options.mode);
runSequence.apply(null, options.sequence.concat([next]));
}
return sequenceTask;
}
module.exports = {
getTask: getSequenceTask,
defaultOptions: {
sequence: []
}
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,803 |
AB Ekmans Mekaniska Snickerifabrik var ett träindustriföretag i Stockholm.
Pehr Johan Ekman grundade 1858 Ekmans Mekaniska Snickerifabrik vid Kungsholmstorg i Stockholm. Efter en brand 1866 uppfördes en ny fabrik vid Barnhusviken. Ekmans snickerifabrik var den största och den äldsta av de fem mekaniserade snickerifabriker som fanns i Stockholm under tillväxtåren på 1880-talet. Med hjälp av en egenkonstruerad trähyvlingsmaskin tillverkade företaget inredningsdetaljer i trä och levererade också från 1860-talet monteringsfärdiga hus, från små paviljonger till villor i olika storlekar, skolhus, kyrkor, stallar och badhus, till den svenska marknaden och på export. Frans Lindskog var verkställande direktör och arkitekt vid företaget 1873-90. Han efterföljdes av arkitekt Carl Olinder 1890–1894. På 1890-talet flyttades fabriken till Lilla Alby.
Ekmans och andra trähustillverkare var pionjärer i Sverige att marknadsföra hus genom illustrerade, tryckta kataloger.
Ekmanska husen, Kungsholmen
Mellan dåvarande repslagargatan och Kungsholmsgatan i kv. Åkermannen uppförde bolaget med start 1877 en hel koloni med arbetarbostäder. 18 trähus i tre rader ingick i anläggningen så ritats av Lindskog. De så kallade Ekmanska husen inrymde bostäder för fyra familjer. Lägenheterna omfattade 1 rum och kök samt tabur och garderob. Gårdarna avgränsades av spjälstaket. På dessa fanns avträdeshus i form av små åttkantiga lushusliknande skapelser samt soplårar. De revs 1897.
Bildexempel, bevarade villor
Bildexempel, rivna byggnader
Se även
Ligna
Referenser
Noter
Källor
Rickard Edlund: Monteringsfärdigt i backspegeln i Rickard Edlund (redaktör): ''Kataloghuset - Det egna hemmet i byggsats, Byggförlaget 2004, läst 2011-10-29
Företag bildade 1858
Industriföretag i Stockholm
Svenska trävaruindustrier
Kungsholmen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,299 |
{"url":"http:\/\/www.2874565.com\/questions\/187070\/regular-unimodular-triangulation-for-a-certain-simplex","text":"# Regular unimodular triangulation for a certain simplex\n\nConsider an $n$-simplex with vertices given by $(0,0,\\dots,r_i,r_{i+1},\\dots,r_n)$ where $r_1,\\dots,r_n$ are given natural numbers, and $i=0,1,\\dots,n+1$.\n\nDoes this simplex admit a regular, unimodular triangulation?\n\n\u2022 Maybe, $r_0$ is also positive integer? \u2013\u00a0Fedor Petrov Apr 8 '16 at 5:25\n\u2022 @FedorPetrov Ah, yes. \u2013\u00a0Per Alexandersson Apr 8 '16 at 12:31\n\u2022 I am a little late to the party, but what is a regular unimodular triangulation? \u2013\u00a0Igor Rivin Apr 3 '17 at 13:41\n\u2022 @IgorRivin I am a little late to answer, but unimodular means onto simplices with integer vertices and volume $1\/n!$; regular means that there exists a convex function on the whole simplex which is affine on each simplex of the triangulation and has different gradients on any two of them. See Gelfand Kapranov Zelevinsky. \u2013\u00a0Fedor Petrov Dec 25 '18 at 4:47\n\nThe answer is yes. Consider the hyperplanes $x_i=x_j+m$ and $x_i=m$ for all $i$ $j$ and integers $m$. These hyperplanes triangulate the entire space into unimodular simplices, and the hyperplanes are compatible with the hyperplanes that determine the polytope in the question (meaning each face is determined by an intersection of some of these hyperplanes). This induces a triangulation of the polytope, and it should be clear that this is regular.","date":"2019-07-18 21:48:21","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\": 0.685996413230896, \"perplexity\": 544.3077760681923}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195525829.33\/warc\/CC-MAIN-20190718211312-20190718233312-00076.warc.gz\"}"} | null | null |
from __future__ import absolute_import, division, print_function
__metaclass__ = type
ANSIBLE_METADATA = {'metadata_version': '1.1',
'status': ['preview'],
'supported_by': 'community'}
DOCUMENTATION = r'''
---
module: purefb_ds
version_added: '2.8'
short_description: Configure FlashBlade Directory Service
description:
- Create or erase directory services configurations. There is no facility
to SSL certificates at this time. Use the FlashBlade GUI for this
additional configuration work.
- To modify an existing directory service configuration you must first delete
an existing configuration and then recreate with new settings.
author:
- Pure Storage Ansible Team (@sdodsley) <pure-ansible-team@purestorage.com>
options:
state:
description:
- Create or delete directory service configuration
default: present
type: str
choices: [ absent, present ]
dstype:
description:
- The type of directory service to work on
choices: [ management, nfs, smb ]
type: str
enable:
description:
- Whether to enable or disable directory service support.
default: false
type: bool
uri:
description:
- A list of up to 30 URIs of the directory servers. Each URI must include
the scheme ldap:// or ldaps:// (for LDAP over SSL), a hostname, and a
domain name or IP address. For example, ldap://ad.company.com configures
the directory service with the hostname "ad" in the domain "company.com"
while specifying the unencrypted LDAP protocol.
type: list
base_dn:
description:
- Sets the base of the Distinguished Name (DN) of the directory service
groups. The base should consist of only Domain Components (DCs). The
base_dn will populate with a default value when a URI is entered by
parsing domain components from the URI. The base DN should specify DC=
for each domain component and multiple DCs should be separated by commas.
required: true
type: str
bind_password:
description:
- Sets the password of the bind_user user name account.
type: str
bind_user:
description:
- Sets the user name that can be used to bind to and query the directory.
- For Active Directory, enter the username - often referred to as
sAMAccountName or User Logon Name - of the account that is used to
perform directory lookups.
- For OpenLDAP, enter the full DN of the user.
type: str
nis_servers:
description:
- A list of up to 30 IP addresses or FQDNs for NIS servers.
- This cannot be used in conjunction with LDAP configurations.
type: list
version_added: 2.9
nis_domain:
description:
- The NIS domain to search
- This cannot be used in conjunction with LDAP configurations.
type: str
version_added: 2.9
join_ou:
description:
- The optional organizational unit (OU) where the machine account
for the directory service will be created.
type: str
version_added: 2.9
extends_documentation_fragment:
- purestorage.fb
'''
EXAMPLES = r'''
- name: Delete existing management directory service
purefb_ds:
dstype: management
state: absent
fb_url: 10.10.10.2
api_token: e31060a7-21fc-e277-6240-25983c6c4592
- name: Create NFS directory service (disabled)
purefb_ds:
dstype: nfs
uri: "ldaps://lab.purestorage.com"
base_dn: "DC=lab,DC=purestorage,DC=com"
bind_user: Administrator
bind_password: password
fb_url: 10.10.10.2
api_token: e31060a7-21fc-e277-6240-25983c6c4592
- name: Enable existing SMB directory service
purefb_ds:
dstypr: smb
enable: true
fb_url: 10.10.10.2
api_token: e31060a7-21fc-e277-6240-25983c6c4592
- name: Disable existing management directory service
purefb_ds:
dstype: management
enable: false
fb_url: 10.10.10.2
api_token: e31060a7-21fc-e277-6240-25983c6c4592
- name: Create NFS directory service (enabled)
purefb_ds:
dstype: nfs
enable: true
uri: "ldaps://lab.purestorage.com"
base_dn: "DC=lab,DC=purestorage,DC=com"
bind_user: Administrator
bind_password: password
fb_url: 10.10.10.2
api_token: e31060a7-21fc-e277-6240-25983c6c4592
'''
RETURN = r'''
'''
NIS_API_VERSION = '1.7'
HAS_PURITY_FB = True
try:
from purity_fb import DirectoryService
except ImportError:
HAS_PURITY_FB = False
from ansible.module_utils.basic import AnsibleModule
from ansible.module_utils.pure import get_blade, purefb_argument_spec
def update_ds(module, blade):
"""Update Directory Service"""
# This module is a place-holder until we figure out a way to
# update the config on the fly rather than deleting and resetting
changed = False
module.exit_json(changed=changed)
def enable_ds(module, blade):
"""Enable Directory Service"""
changed = True
if not module.check_mode:
try:
blade.directory_services.update_directory_services(names=[module.params['dstype']],
directory_service=DirectoryService(enabled=True))
changed = True
except Exception:
module.fail_json(msg='Enable {0} Directory Service failed: Check Configuration'.format(module.params['dstype']))
module.exit_json(changed=changed)
def disable_ds(module, blade):
"""Disable Directory Service"""
changed = True
if not module.check_mode:
try:
blade.directory_services.update_directory_services(names=[module.params['dstype']],
directory_service=DirectoryService(enabled=False))
except Exception:
module.fail_json(msg='Disable {0} Directory Service failed'.format(module.params['dstype']))
module.exit_json(changed=changed)
def delete_ds(module, blade):
"""Delete Directory Service"""
changed = True
if not module.check_mode:
dirserv = blade.directory_services.list_directory_services(names=[module.params['dstype']])
try:
if module.params['dstype'] == 'management':
if dirserv.items[0].uris:
dir_service = DirectoryService(uris=[''],
base_dn="",
bind_user="",
bind_password="",
enabled=False)
else:
changed = False
elif module.params['dstype'] == 'smb':
if dirserv.items[0].uris:
smb_attrs = {'join_ou': ''}
dir_service = DirectoryService(uris=[''],
base_dn='',
bind_user='',
bind_password='',
smb=smb_attrs,
enabled=False)
else:
changed = False
elif module.params['dstype'] == 'nfs':
if dirserv.items[0].uris:
dir_service = DirectoryService(uris=[''],
base_dn='',
bind_user='',
bind_password='',
enabled=False)
elif dirserv.items[0].nfs.nis_domains:
nfs_attrs = {'nis_domains': [],
'nis_servers': []}
dir_service = DirectoryService(nfs=nfs_attrs,
enabled=False)
else:
changed = False
if changed:
blade.directory_services.update_directory_services(names=[module.params['dstype']],
directory_service=dir_service)
except Exception:
module.fail_json(msg='Delete {0} Directory Service failed'.format(module.params['dstype']))
module.exit_json(changed=changed)
def create_ds(module, blade):
"""Create Directory Service"""
changed = True
if not module.check_mode:
try:
if module.params['dstype'] == 'management':
if module.params['uri']:
dir_service = DirectoryService(uris=module.params['uri'][0:30],
base_dn=module.params['base_dn'],
bind_user=module.params['bind_user'],
bind_password=module.params['bind_password'],
enabled=module.params['enable'])
else:
module.fail_json(msg="URI and associated params must be specified to create dstype {0}".format(module.params['dstype']))
elif module.params['dstype'] == 'smb':
if module.params['uri']:
smb_attrs = {'join_ou': module.params['join_ou']}
dir_service = DirectoryService(uris=module.params['uri'][0:30],
base_dn=module.params['base_dn'],
bind_user=module.params['bind_user'],
bind_password=module.params['bind_password'],
smb=smb_attrs,
enabled=module.params['enable'])
else:
module.fail_json(msg="URI and associated params must be specified to create dstype {0}".format(module.params['dstype']))
elif module.params['dstype'] == 'nfs':
if module.params['nis_domain']:
nfs_attrs = {'nis_domains': [module.params['nis_domain']],
'nis_servers': module.params['nis_servers'][0:30]}
dir_service = DirectoryService(nfs=nfs_attrs,
enabled=module.params['enable'])
else:
dir_service = DirectoryService(uris=module.params['uri'][0:30],
base_dn=module.params['base_dn'],
bind_user=module.params['bind_user'],
bind_password=module.params['bind_password'],
enabled=module.params['enable'])
blade.directory_services.update_directory_services(names=[module.params['dstype']],
directory_service=dir_service)
except Exception:
module.fail_json(msg='Create {0} Directory Service failed: Check configuration'.format(module.params['dstype']))
module.exit_json(changed=changed)
def main():
argument_spec = purefb_argument_spec()
argument_spec.update(dict(
uri=dict(type='list'),
dstype=dict(required=True, type='str', choices=['management', 'nfs', 'smb']),
state=dict(type='str', default='present', choices=['absent', 'present']),
enable=dict(type='bool', default=False),
bind_password=dict(type='str', no_log=True),
bind_user=dict(type='str'),
base_dn=dict(type='str'),
join_ou=dict(type='str'),
nis_domain=dict(type='str'),
nis_servers=dict(type='list'),
))
required_together = [['uri', 'bind_password', 'bind_user', 'base_dn'],
['nis_servers', 'nis_domain'],
['join_ou', 'uri']]
mutually_exclusive = [['uri', 'nis_domain']]
module = AnsibleModule(argument_spec,
required_together=required_together,
mutually_exclusive=mutually_exclusive,
supports_check_mode=True)
if not HAS_PURITY_FB:
module.fail_json(msg='purity_fb sdk is required for this module')
state = module.params['state']
blade = get_blade(module)
api_version = blade.api_version.list_versions().versions
ds_configured = False
dirserv = blade.directory_services.list_directory_services(names=[module.params['dstype']])
ds_enabled = dirserv.items[0].enabled
if dirserv.items[0].base_dn is not None:
ds_configured = True
if (module.params['nis_domain'] or module.params['join_ou']) and (NIS_API_VERSION not in api_version):
module.fail_json(msg="NFS or SMB directory service attributes are not supported in your FlashBlade Purity version.")
ldap_uri = False
set_ldap = False
for uri in range(0, len(dirserv.items[0].uris)):
if "ldap" in dirserv.items[0].uris[uri].lower():
ldap_uri = True
if module.params['uri']:
for uri in range(0, len(module.params['uri'])):
if "ldap" in module.params['uri'][uri].lower():
set_ldap = True
if not module.params['uri'] and ldap_uri or \
module.params['uri'] and set_ldap:
if module.params['nis_servers'] or module.params['nis_domain']:
module.fail_json(msg="NIS configuration not supported in an LDAP environment")
if state == 'absent':
delete_ds(module, blade)
elif ds_configured and module.params['enable'] and ds_enabled:
update_ds(module, blade)
elif ds_configured and not module.params['enable'] and ds_enabled:
disable_ds(module, blade)
elif ds_configured and module.params['enable'] and not ds_enabled:
enable_ds(module, blade)
# Now we have enabled the DS lets make sure there aren't any new updates...
update_ds(module, blade)
elif not ds_configured and state == 'present':
create_ds(module, blade)
else:
module.exit_json(changed=False)
if __name__ == '__main__':
main()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,250 |
Homing's In: events Nov. 5-14
by Jim Provenzano
Thursday Nov 5, 2020
Bearracuda, Trans FilmFestival, Tina D'Elia
Well, that was something. Is it over yet? You need some arts, nightlife and community gatherings, online and in-person, to calm down and relax.
Author Events at Alley Cat Books
The bookstore, while physically closed, hosts online events with acclaimed authors. the store is also taking mail order book purchases. www.alleycatbookshop.com
Author Events at Booksmith
The popular Upper Haight bookstore hosts talks and readings with mainstream fiction, nonfiction authors. https://www.booksmith.com
Author Events at Bureau of General Services-Queer Division
Enjoy literary events hosted by the independent bookstores at the NYC LGBT Center. Nov 6,
3pm PT: Anais Duplan and Joshua Escobar. Nov 12, 3pm PT, authors Felice Picano and Jim Provenzano discuss their two new Hollywood-themed novels. Free. www.facebook.com event page www.eventbrite.com https://bgsqd.com/
Author Events at City Lights
the historic bookstore presents online author events. http://www.citylights.com/
Author Events at Green Apple Books
Online readings and chats; in-store shopping. Nov. 13, 7pm: Patrick Early Ryan and Martin Pousson. Nov. 18, 5pm: Alden Jones and Rick Moody. www.greenapplebooks.com
authors Jim Provenzano and Felice Picano at a Bureua of General Services-Queer Division online event
Odd Mondays
Chia-Chia Lin, Cara Black, and Wayne Goodman read about immigrant Alaska, WWII France, and Queer 17th Century Japan at "Faraway Places." Nov. 9, 7pm on Zoom. RSVP: oddmondaysnoevalley@gmail.com
QueerWords Podcast
Wayne Goodman's online audio discussions with many notable LGBTQ fiction, poetry authors, essayists and journalists. https://queerwords.org/
American Pavilion Films
Virtual debut for a diverse showcase of 38 incredible new films between all six of the showcases — Student Short Films, Student Documentaries, Emerging Filmmaker Short Films, Emerging Filmmaker Documentaries, High School Showcase, and Emerging Filmmaker LGBTQ Showcase Films. https://www.ampav.com/
Quirky, anxiety-fueled gay rom-com directed by Simon (Nevermind the Buzzcocks) Amstell and starring Colin Morgan (Merlin, The Happy Prince), is now available on DVD and digital platforms including Kino Now.
Bloodsisters: Leather, Dykes And Sadomasochism
25th anniversary online screenings of the historic documentary, with Michelle Handelman
Featuring Tala Brandeis, Patrick Califia, J.C. Collins, Queen Cougar, Amy Marie Meek, Skeeter, Wickie Stamps, Peggy Sue, Ann Soucy-West & others. https://newfest.org/event/bloodsisters/
Jonah Greenstein's feature debut about inter-related straight and gay people in a disturbed rural Iowa town; viewable on Apple TV, Amazon Prime, OVID.tv & Kanopy. http://www.firstrunfeatures.com/intheaters.html
11th edition of the country's largest documentary film festival, with several LGBT-themed films: Bare, Can You Bring It: Bill T. Jones and D'Man in the Waters, No Ordinary Man, Origin of the Species and other films. Nov 11-19. https://www.docnyc.net/
The Garden Left Behind
Winner of SXSW's Audience Award and named one of the best LGBTQ films of all time by Rotten Tomatoes, Flavio Alves's moving drama of an undocumented Mexican trans woman in New York City can be viewed online. https://www.cinematropical.com/
New HBO Max series of docudrama about prominent figures in LGBT history, with Billy Porter, Cheyenne Jackson, Anthony Rapp, Sara Gilbert, Anne Ramsay and others. https://www.hbomax.com
Films at The Roxie
Watch independent and classic films online at the Roxie's virtual cinema.
https://www.roxie.com
'the Garden Left Behind'
GayBingeTV
Streaming channel for independent LGBT short, feature and documentary films.
https://www.gaybingetv.com/
Named to Indiewire's "Fabulous LGBTQ Films Not To Miss," Holy Trinity is the story of a sex-positive dominatrix in Chicago who discovers a newfound (and drug-induced) talent for speaking to the dead. Molly Hewitt wrote, directed and stars in this wild ride of queer spirituality and alternative sexuality. https://www.fullspectrumfeatures.com/trinity
Homo Say What
Acclaimed documentary explores the origins of homophobia against men that became institutionalized in the 1950s and how that continues to negatively affect all genders and sexualities to this day. https://homosaywhatfilm.com/
House of Cardin
The acclaimed documentary about fashion designer Pierre Cardin's online. http://houseofcardin.com/
LGBTQ Films on Revry
The global queer streaming network screens free and subscription-paid feature films, documentaries, short films and episodic series. New: OML, the 27.7 women's channel, featuring a new series, Socially Distant. https://revry.tv/
LGBTQ Films on Tubi
The free streaming service showcases several queer films, including God's Own Country (2017) - won directing award at Sundance; Hearthstone (2016) — won directing award at TIFF; Jeffrey (1995) — starring Patrick Stewart; Latter Days (2003); Lip Service (2012, series); Love is All You Need? (2016) — starring Emily Osment; Poison (1991) - directed by Todd Haynes, Sundance Grand Jury Prize winner; Saved! (2004) — starring Mandy Moore and Macaulay Culkin, and others. https://tubitv.com/
New documentary about Ellis Haizlip, the 1970s Black TV host who brought Black actors, activists, musicians, and community leaders to TV screens. https://www.mrsoulmovie.com/
Oakland International Film Festival
Rent VOD feature, documentary and short films. https://theoaklandinternationalfilmfestival.vhx.tv/products
'Old Show Queens'
Old Show Queens
Amusing insightful interview series with gay couple Gary Gunas and Billy Rosenfield, super and and producers of Broadway shows. https://www.oldshowqueens.com/
Oliver Sacks: His Own Life
Fascinating documentary about the gay psychotherapist and author. https://www.oliversacksdoc.com/
OutFest 2020
View LGBT films and artist discussion panels online. https://outfest.org/alwayson/
Prideland
New interview series; Out, Proud & Southern: Dyllón Burnside's Story; recounts his own experience coming out in the South and travels to the heart of Alabama to meet Carmarion D. Anderson, a barrier-breaking transgender woman of color and the Alabama State Director of the Human Rights Campaign. www.pbs.org
Promo Homo TV
Veteran journalist Nicholas Snow hosts an online series of video interviews with LGBT actors, directors, authors and community activists. https://promohomo.tv
Philippe Faucon's three-part dramatic film about French gay lives in 1981, 1999 and 2013. https://kinomarquee.com/
Putting On
Season 2 of the gay reality show airs (starring Israeli-born underwear model On Mekahel); Sundays at 4pm & 7pm. https://revry.tv/
Queen of the Capital
Josh Davidsburg's vivacious and inspiring documentary follows Muffy Blake Stephyns, a drag performer with high aspirations. Muffy and her everyday persona, Daniel, navigate the drag and bureaucratic worlds of Washington, D.C., while following her dream to lead a colorful group of philanthropic drag performers. $10. https://ondemand.drafthouse.com/film/queen-of-the-capital/
Queer East Film Festival
Mini-festival of LGBT Asian films. https://queereast.org.uk/
'Sideways Smile'
Sideways Smile
New comedy web series about a young queer Asian American woman's urban journey of sexual self-discovery. https://sidewayssmileseries.com/
Jason Stuart and Mitch Hara's two-man comedy film, now on Amazon Prime, YouTube, Revry. https://www.smotheredtv.com/
Indie film dramatizes modern day political divides in the LGBTQ Community, made by writer, director and star, Jack Tracy. https://vimeo.com/ondemand/snowflakefilm
Fascinating documentary film about gay-run Mardi Gras crews, from their early years to today. https://vimeo.com/ondemand/sonsoftennesseewilliams
Terrence McNally: Every Act of Life
Screening of the American Masters documentary about the the late gay playwright. http://www.pbs.org/wnet/americanmasters
Transgender Film Festival
20th annual festival with seven programs of short films with a variety of cinematic styles; All films are Closed Captioned for Deaf and Hard-of-Hearing audiences. Nov. 12-15. http://sftff.org/
Virtual Streaming Room With Cinema SF
Cinema SF, owners of the historic Balboa and Vogue Theaters, are keeping the spirit of movie-going alive with a series of virtual movie screenings and online chats. Proceeds from the screenings will go towards keeping the theaters alive so they can reopen after the shutdown ends. https://www.cinemasf.com/
Where the Bears Are
New online comedy show with two mature gay men. https://youtu.be/5TQnRDc15i0
Animals worldwide have reclaimed urban spaces, as you may have seen in the news. But we need to face how our damage to the environment and abuse of wild animals may have aided the spread of disease. While we contemplate that, enjoy films of animals in the wild. Peninsula Open Space Trust (POST) hosts an ongoing series of nature films and talks. openspacetrust.org
Cal Performances at Home
Watch videos of classical and modern music concerts, and artist conversations courtesy of the Berkeley-based performing arts organization. $5-$60 and up. Full new season schedule at www.calperformances.org
Garrin Benfield
Enjoy live and recorded home sets performed by the gay singer-composer. Garrin's YouTube channel
The Klipptones
Local jazz-pop band performs weekly streaming concerts; Fridays, Sundays and Mondays. https://www.facebook.com/klipptones/live
Noe Music
Online concerts in many genres (classical, folk, early music). Nov 8, 7pm: Third Coast Percussion. https://www.noemusic.org/
Old First Concerts
Enjoy live streaming and archived music concerts of classic, and contemporary music. https://www.oldfirstconcerts.org
Online screenings of the Fleetwood Mac singer's '24 Karat Gold' solo concerts. https://www.stevienicksfilm.com/
San Francisco Gay Men's Chorus
Enjoy concerts and films, plus guest performers joining their Behind the Curtain series and Summer Reruns (Britney Coleman, Lisa Vroman, Billy Porter, Wilson Cruz, Adam Rippon, Britney Coleman, Laura Benanti, Andrew Lippa); plus Chasten and Pete Buttigieg, Wanda Sykes, Sharon Stone, Martha Wash and other performers. https://www.sfgmc.org/sfgmctv
San Francisco Performances
After several programs in celebration of their 40th anniversary were cancelled, instead, SFP offers memorable classical concerts from the past, all recorded at the Herbst Theater between 2004-2008. Concerts are audio only. http://sfperformances.org/
Seth Concert Series
Broadway, film and TV actor-singers perform intimate concerts online with pianist and musical theatre expert Seth Rudetsky. $20-$25. Nov. 8 & 9: Jessie Mueller (Tony Award winner). Nov. 8: Wayne Brady. Nov. 15 & 16: Lillias White. https://thesethconcertseries.com/
SF Jazz Fridays at Five
Jazz concerts will bring you good vibes. The acclaimed venue brings their concert series online with talented musicians. https://www.sfjazz.org
Susan Werner
The accomplished singer-songwriter (Bull Durham, The Musical) performs acoustic songs with friends online, including music from her new album, 'Flyover Country' series in support of the Joe Biden/Kamala Harris campaign. http://susanwerner.com/
'Art' at SF Playhouse
THEATER, DANCE, OPERA
Amy Siewert: Imagery
Dance is the most vibrantly physical of art forms, and we miss it. Until we can enjoy it live, check out some amazing short dance films by Siewert's beautiful deft dancers. The tenth anniversary Sketch series presents Red Thread, a new documentary dance film project with four films online. Nov 5. 6:30pm PT; Premiere of Purple Skin by Stephanie Martinez and Dean Berdusis,www.asimagery.org https://www.eventbrite.com
Produced by San Francisco Playhouse. Video on-demand. $15-$100. This brand new, fully staged production of Yasmina Reza's bitingly funny take on taste, values and modern art was mounted for video recording just last week at SF Playhouse's Post Street theater. Directed by Bill English, this contemporary favorite's cast of three includes Jomar Tagatac, one of the Bay Area's most versatile actors, always a welcome presence on stage...and now in our homes. http://bit.ly/ArtBAR
Blood at the Root
Dominique Morriseau's ensemble drama based on the Jena Six: the six Black students who were initially charged with attempted murder for a school fight after being provoked with nooses hanging from a tree on campus. Live Nov 6, 7:30pm. On-deman Nov 7-15. http://therhino.org
Broadway on AirBnB
Promotional-sponsored performances by a variety of talents: Tittus Burgess, cast members from Moulin Rouge, Mrs. Doubtfire, and more, plus dance, yoga, tarot and circus classes. $10-$50. www.airbnb.com
Broadway Performers
For Broadway fans, Tony-winning performers perform new concerts, and classic shows are streamed as well. https://www.broadwayworld.com/
Musical show with a current theme: How do these emotions collide during time spent alone in quarantine? When James opens a box of artifacts from his first breakup, after months of self-isolation during COVID-19, he reckons with memories that haunt him of a first queer love gone awry; starring James Jackson, Jr., Victoria Huston-Elem, Danielle Buonaiuto, and Karl Saint Lucy. https://www.differentstars.live/
Flamingos Forever
"A radio play in poor taste," a new script inspired by the John Waters classic camp film Pink Flamingos, where The Queen of Filth and her family return to Baltimore for the first time since 1972. $10 gets a full download plus other goodies. https://www.queerpowers.com/homodaze-2
The Flats, produced by the Aurora Theatre Company; audio serial on-demand. The three episodes of this highly anticipated collaboration between top Bay Area playwrights Lauren Gunderson, Cleavon Smith and Jonathan Spector will be released on three consecutive Fridays, then each will remain available for an open-ended streaming run. A single ticket covers all three segments, which focus on the diverse residents of a Berkeley triplex who find themselves sheltering-in-place due to a mysterious threat. $20. www.auroratheatre.org
From Broadway to Hollywood
42nd Street Moon's online streams of various shows. https://42ndstmoon.org/
The Golden Girls, a Puppet Parody
The favorite sitcom is performed by puppets, directed by Jonathan Rockefeller; 16 episodes. https://thatgoldengirlsshow.com/
New Conservatory Theatre Center's new podcast drama-comedy, with playwrights and producers dealing with COVID in a realistic story. https://www.nctcsf.org/In-Good-Company-Podcast
Produced by the Berkeley Rep; audio on demand. Free. Four years ago, the Berkeley Rep mounted its stage adaptation of Sinclair Lewis' satirical 1935 novel about a fascist demagogue, Buzz Windrip, who becomes president of the United States. Recorded live on October 13, the four episodes are now posted for streaming at http://bit.ly/CantHappenHereBAR
Josh Kornbluth's 'Citizen Brain'
Josh Kornbluth's Citizen Brain
Produced by Shotgun Players, acclaimed Bay Area-based comic monologuist Kornbluth draws on his 2016 artist's residency at UCSF's Global Brain Health Institute to creatively delve into connections he sees between dementia and the current American political system. Live on Zoom at 7pm. Thursdays through Saturdays and 5pm on Sundays. http://bit.ly/CitizenBrainBAR
The Law of Attraction
New Conservatory Theatre Center's world premiere of Patricia Milton's adapted radio comedy about bad manners. Thru Nov. 18. Free/$40. https://www.nctcsf.org/
Live at The Lortel
Season 2 of the podcast series, with musical theatre talents discussing their art and lives. Archived podcasts include BD Wong. liveatthelortel.com
Los Altos Stage Company
Fundraising performance of Harold Pinter's Old Timed with Brooke Adams, Polly Draper and Tony Shalhoub. Nov. 8, 5pm PT. www.losaltosstage.org
The Marsh Stream
The theatre producer known for acclaimed solo shows presents a variety of programming at 7:30pm nightly on MarshStream, its newly launched Broadcast Platform, even interactive Bingo game nights. http://themarsh.org/marshstream/
New Conservatory Theater Company: Offstage But Online
Temporarily shuttered, queer theater company NCTC is now building a digital playground for your viewing pleasure that will feature the new podcast drama, actor monologues, messages from playwrights, exercises to practice mindfulness and original material from Conservatory students. Quarantine is a Drag Trivia Night, with J. Conrad Frank (Katya Smirnoff-Skyy).
https://www.nctcsf.org/onstage-to-online
www.facebook.com/nctcsf
https://www.youtube.com/user/NCTCSF
Overlooked Latinas
Tina D'Elia's solo show about a butch lesbian TV star and a femme fatale Nov. 13-22. http://therhino.org
Berkeley Repertory Theatre presents a bunch of short play scripts you can download and perform at home, featuring works by Julia Cho, Peter Sinn Nachtrieb, Miranda Rose Hall, Ricardo Pérez González and several others. www.playathome.org
Live and recorded solo and group plays and performances from the acclaimed New York City theater company; Under the Greenwood Tree, a new documentary about the 2017 production of As You Like It. https://publictheater.org/
Tina D'Elia's 'Overlooked Latinas'
PUSHfest
Online streaming international collection of dances; choreographer, activist and community organizer Raissa Simpson's PUSH Dance Company continues its 15th anniversary year by increasing its programming by presenting PUSHFest Global, an online streaming festival of current and contemporary dance, with 18 Bay Area dancemakers and companies, and a few from beyond the Bay, in seven live, online presentations every Wednesday through Nov. 18. https://www.pushdance.org/festival
SF BATCO
San Francisco Bay Area Theatre Company streams weekly new plays, readings, arts talks, improv and more. https://www.twitch.tv/sfbatco
SF Opera
Stream recordings of the company's performances of classic operas. Nov 7, 1am PT: Verdi's tuneful masterpiece, A Masked Ball. https://sfopera.com
SF Playhouse Zoomlets
the acclaimed SF theatre company adapts, with short online plays. Also, weekly playwright chats. Ticketed videos on-demand include Brian Copeland's The Jewelry Box, From Blues the Broadway (collaborative music play), and Art by Yasmina Reza. Free/donations. https://www.sfplayhouse.org
Smuin Contemporary Ballet
The local dance company streams notable works, including Helen Pickett's Oasis, Brennan Wall's Nocturne, Rex Wheeler's Sinfonietta and an ongoing Hump Day Ballets series. https://www.smuinballet.org/
Stars in the House
Musical savant and affable host Seth Rudetsky —with his husband, producer James Wesley— hosts Zoom chats with Broadway, film and TV stars galore. One of the first online fundraisers (for The Actor's Fund), your donations help artists of all kinds, from actors to lighting crew members. Broadway may be closed, but you can enjoy living room concerts and chats with favorite stars. www.starsinthehouse.com
Theatre Rhino
The local gay theatre company presents online performances of solo and other plays, including works (mostly) written and performed by John Fisher. Nov. 5: Insane Director. Nov. 10, 7pm: Terry Guest's At the Wake of a Dead Drag Queen, performed by Charles Peoples III and Mario Mazzetti. All plays archived online. www.therhino.org
We Players' Audio Dramas
the innovative site-specific theatre company presents podcast-style audio re-interpretations of their recent productions, including Wonder Storytime with Alice and Friends, and The Odyssey. www.weplayers.org
WQUR: Queer Quarantine Radio
BenDeLaCreme, Jinkx Monsoon, Peaches Christ and Major Scales perform old-time radio-style thriller detective programs. https://anchor.fm/wqur
The great thing about the big museums is they have a lot of space, so social distancing is easy. Don your fashionable mask and peruse the reopened exhibits of contemporary and ancient works at the Asian Art Museum. Exhibits include Awaken: A Tibetan Buddhist Journey Toward Enlightenment, Jenifer K Wofford: Pattern Recognition, a mural that celebrates Asian American artists, Lost at Sea: Art Recovered from Shipwrecks, and Chang Dai-chien: Painting from Heart to Hand. Among the outdoor exhibits is Don't Mess With Me, a vibrant colorful mural by Mumbai-based street artist Jas Charanjiva. Fri-Mon 10am-5pm. Thu 1pm-8pm. 200 Larkin St. asianart.org
Berkeley Art Museum/Pacific Film Archive
Closed to the public for now, view online exhibits (Ron Nagle: Handsome Drifter, Folk and Modern Art in Postcolonial India) and art films streamed online. 2155 Center St., Berkeley. www.bampfa.org
Reopened to the public (masks and social distance requirements), the fascinating science museum includes live creatures (aquarium, terrarium) and educational exhibits. Mon-Sat 9:30am-5pm. Sun 11am-5pm. 55 Music Concourse Drive, Golden Gate Park. https://www.calacademy.org/
Catharine Clark Gallery
Politics takes center stage at well-timed exhibits of works by Nina Katchadourian (Monument to the Unelected) and To the Person Sitting in Darkness (four artists' focus on disenfranchised citizens, including Stephanie Syjuco's stark interpretation of a toxic U.S. flag); also, online exhibits of regional artists' works. 248 Utah St. cclarkgallery.com
Chabot Space & Science Center
Online programs (the center is closed to the public) include live science programs each Wed., 10:30am, virtual telescope viewings each Sat. 9pm. chabotspace.org
Contemporary Jewish Museum Events & Exhibits
Reopened to the public, the spacious downtown museum includes Stephen Berkman's Predicting the Past: Zohar Studios, The Lost Years, an immersive photography installation about a mythical 19th-century Jewish photographer. Levi Strauss: A History of American Style continues through Jan. 10, 2021; Threads of Jewish Life: Ritual and Other Textiles from the San Francisco Bay Area continues through Feb, 28, 2021. Free/$16. 736 Mission St. thecjm.org
The stylish modern museum located in the heart of Golden Gate Park has reopened. Major exhibits include Friday Kahlo: Appearances Can Be Deceiving (through Feb. 7, 2021), Uncanny Valley: Being Human in the Age of AI (through June 26, 2021), collections of American, African, Oceanic, costume arts and more; and a special Oct. 28 event, [Virtual] Local Voices Live: Marilet Martinez and Celia Sagastume. Tue-Sun 9:30am-5:15pm (extended to 7:45pm, Fridays and Saturdays through November for the Frida Kahlo exhibit). 50 Hagiwara Tea Garden Drive. deyoung.famsf.org
Exploratorium Events
Talks, and demos with a science theme, including comet-cam and more. Nov. 12, 7pm: After Dark online covers 'animal Intelligence.' www.exploratorium.edu
GLBT Historical Society events & exhibits
The museum reopened Oct. 1, with advance reservations required. Fighting Back, a series that brings together community leaders, experts, historians and activists to explore lessons from the past that might be useful in formulating resistance efforts today. Online exhibits include Reigning Queens: the Lost Photos of Roz Joseph, curated by Joseph Plaster; Angela Davis: OUTspoken, Labor of Love about the first ten years of Pride events in San Francisco; Performance, Protest & Politics: The Art of Gilbert Baker, Pioneering Periodicals, 1940s-1950s, Picturing Kinship: Portraits of our Community by Lenore Chinn and 50 Years of Pride. November programs focus on queer history podcasts (Nov. 6), lesbian artists defying Nazis in Germany (Paper Bullets, Nov 13). $5 donation. 4127 18th St. www.glbthistory.org
Gregangelo Museum
Take online and in-person tours (using all safety protocols with 'The Riddle of the Sphinx,' a new outdoor show) at the 27-room mansion filled with eccentric unusual interactive artworks and performances. www.gregangelomuseum.com
SOMArts Cultural Center
For classic art set in a stately building with an ocean view, nothing beats the Legion of Honor. European and ancient art, giant Renaissance landscapes and historic paintings, plus classical sculptures and contemporary works. From Degas, Manet, Caillebotte and Vuillard to Egyptian, Greek and Roman artifacts, the various collections will leave you awe-struck. Tue-Sun, 9:30am-5:15pm. Free/$15. legionofhonor.famsf.org
Mercury 20 Gallery
It's never too early to start gift shopping. Many galleries offer online purchase options, and you can have a print, painting, or textile shipped without even leaving home. Mercury 20 Gallery does just that, with a compelling array of colorful works of art. Exhibits include works by Andrea Brewster, Tara Esperanza, Fernando Reyes. Notable sale works include Chris Komater's sexy nudes of gay bear Jack Radcliffe, and Johanna Poethig's timely Covid Spring (2020), an archival print (15 x 20 inches) for $180. 475 25th St., Oakland. Fri-Sat 12pm-6pm. mercurytwenty.com
Mission Cultural Center for Latino Arts
Although still closed to the public, the Center offers online virtual tours and Day of the Dead activities, poetry-reading videos, music and dance concerts and workshops. 2869 Mission St. missionculturalcenter.org
MOAD Film Club
Museum of the African Diaspora screens award-winning films, with multiple talks and music performances online as well. https://www.moadsf.org/
NIAD Exhibits
The art studio for developmentally disabled adults shares online exhibits and sells beautiful works of art. www.niadartstore.org
Oakland Museum
The East Bay museum remains closed, but online exhibits include You Are Here: California Stories on the Map (through 2022), Black Power and Dorothea Lange: Photography as Activism (both ongoing). 1000 Oak St., Oakland. museumca.org
Nancy Toomey Fine Art
What Kind of Cool (Will We Think of Next), an exhibit of collaborative works by the two multimedia artists Rodney Ewing, Tahiti Pehrson. thru Nov 30, by appointment only. 1275 Minnesota St. nancytoomeyfineart.com
SF Museum of Modern Art
Pop, Abstract, and Figurative art collections include some easily recognizable famous works, even for an art-loving amateur. Warhol, Rosenquist, Lichtenstein, to name a few. But give yourself a day to enjoy all five floors of visual treats, since the massive redesign of the building makes for an easy journey through the past century of compelling works. New exhibits include David Park: A Retrospective (through Jan. 2021) and Bay Area Walls, a group mural project in response to 2020's chaotic events. Don't forget the store to get a jump on your holiday shopping. Free/$25. Mon 10am-5pm. Thu 1pm-8pm. Fri-Sun 10am-5pm. 151 Third St. www.sfmoma.org
Living Legacies: Dia de Los Muertos 2020, the 21st annual Day of the Dead exhibit, includes a large group exhibit curated by Rio Yañez and Carolina Quintanilla. Open to the public, timed ticketed entry. $10. Thru Nov. 6. Tue-Fri 112pm-7pm Sat 12pm-5pm. 934 Brannan St. somarts.org
Walt Disney Family Museum
Located in The Presidio, the museum dedicated to the art of Disney films and designs re-opens Nov. 5, with online exhibits and events, including the Happy Haunts Tour. 104 Montgomery St. www.waltdisney.org
Bearracuda
Virtual party for bears, cubs and fans, with DJs Mateo Degade, Feddy Kop, Christeene and Matt Barracuda cohosting a virtual butt contest. Nov. 7, 9pm-12am. $10. http://bearracuda.com/
Beaux-Tanical Gardens
Beaux's extra-spacious outdoor dining and cocktails series delicious food and drinks. Mob-Wed 3pm-9pm. The 3pm-10pm. Fri & Sat 12pm-11pm. Sun 12pm-10pm. https://www.facebook.com/beauxsf
Bootie Mashup
Adriana A and guest DJs blend pop hits to danceable grooves in livestreams, Tuesdays-Saturdays. https://www.twitch.tv/BootieMashup with special October mixes and Halloween 'Monster Mashup' sets Oct. 30 & 31. Also, DNA Lounge weekly webcasts (and DNA Pizza's open for take-out). https://www.dnalounge.com/
Boy Division
DJ Xander hosts the Twitch online edition of the New Wave retro-fun dance party. www.twitch.tv/deejayxander
Brunch with Poppy
Enjoy outdoor patio brunch and a drag show hosted by Poppy; $16 bottomless mimosas. Sundays, 11am-3:30pm. Jones, 620 Jones St. https://www.eventbrite.com
Digital Pride Fest
Enjoy drag and cabaret performances with Tom Goss, Bright Light, Bright Light, Divina De Campo, Trinity the Tuck, Pearl & Precious Brady-Davis, Deven Greeen & Handsome Ned, Miz Cracker, Desmond is Amazing, Manila Luzon and many more. www.digitialpridefest.com
Drag Alive from The Stud
Every Saturday night The Stud, San Francisco's oldest queer bar, will host Drag Alive. Vivvyanne Forevermore and Jillian Gnarling bring you a virtual Stud to help support the staff, performers and the goals for a future venue. Party with the crew of The Stud at Twitch TV: https://www.twitch.tv/dragalive
Drag Queen Mukbang
Online episodic mini-series of drag performers in a show sponsored by Jack Daniel's new brand, Tennessee Fire, with Patrick Starr, Gia Gunn, Eugene Lee Yang and Laganja Estranja. Saturdays. www.jackdaniels.com
Frolic Quarantine
The fun pursuit party goes virtual. Don your animal gear and enjoy a variety of DJ mixes. Monthly, first Saturdays. http://frolicparty.com/
Patti LaBelle at the Mighty Real Gala
Mighty Real Gala
Patti LaBelle is an honored guest, and will sing, at Positive Resource Center's fundraiser gala. Nov. 7, 6pm-7:15. $100 and up; optional dinner and wine from Absinthe delivered to your door. prcsf.org/MightyReal
The Monster Show
One of the longest running drag shows in the Castro, The Monster Show has made The Edge bar a Thursday night destination. The Monster show now comes to Twitch TV, where the performers will entertain you while they raise money for bar employees. https://www.twitch.tv/monstershowsf
Oasis TV
The SoMa nightclub's events include in-person rooftop and outdoor/parklet street events (Thursday-Saturday), online new and archival footage from the venue's best shows; The Golden Girls Live. Cyberotica: the Rock Musical, Sex and the City Live, Buffy the Vampire Slayer Live, AbFab Live, Star Trek Live, Above and Beyond the Valley of the Ultra Showgirls, Debbie Does Dallas, Bitch Slap. Also, Meals on Heels, the hit drag dinner delivery service. sfoasis.com/
Glamamore's long-running Monday night party at Powerhouse can also be found at Twitch TV. The party is called Pillows and has been a staple of the bar for years and is currently being hosted by Mary Vice. It's better than Monday Night Football! Check it out every Monday at 9pm. https://www.twitch.tv/cashmonetdrag
Quarant-Tea
One of the first local nightlife fundraisers, the SF Queer Nightlife Fund presents DJs spinning hours of music for your now possibly obsessive housecleaning activities. https://www.twitch.tv/sfqueernightlifefund
Quarantinis with Katya
Katya Smirnoff-Skyy sings and hosts weekly Thursday online cocktail parties. www.facebook.com/katya.smirnoffskyy
Queerpowers.com
A new LGBTQ+ platform developed by two San Franciscans on the first day of the Covid-19 quarantine. enjoy live drag shows, queer house parties, political broadcasts and other LGBTQ content from all over the world, uploaded hourly. Founded and curated by former Google employee Katie Bush and former Apple employee Anders Howerton. https://www.queerpowers.com/
Red Hots Burlesque
enjoy drag and strip shows, music acts and more in online performances.
http://redhotsburlesque.com/
Russell Deason's Happy Hour in Exile
The Bestie-winning pianist and cabaret singer brings his martini bar ambiance into your home; Mondays 5:30pm-7pm. www.facebook.com
SF Eagle
The famed leather bar now features wacky karaoke, DJs and other entertainments. https://www.twitch.tv/sfeaglebar
Sampson McCormick
The acclaimed comic shares insightful witty perspectives of a Black gay man in short clips and "Church Boy," a new 40-minute recorded show. http://www.sampsoncomedy.com
Sundance Saloon Online Live
Sundance Saloon offers online dance lessons and dancing every Sunday night. Beginning classes are aimed at novices. Learn how to dance while you shelter in place from 6-9pm. 6pm: beginning line dance for complete novices; 7pm: intermediate line dance; 8:15pm: line dance mini-marathon. http://www.sundancesaloon.org/
Sunday Funday, Events at Port Bar Oakland
Yes, the bars with tasty cocktails and limited to-go and dine-out dishes is open. Sunday 12pm-8pm. Fri & Sat 3pm-12am. Tuesdays Big Gay Trivia, 8pm. Drag Queen Bingo, outside, hosted by Vicodonia Nightingale. Wed. night outdoor drag shows, at 7pm, 9pm, 11pm. Sunday Drag Brunch with BeBe Sweetbriar, Delilah BeFierce. 2023 Broadway, Oakland. https://www.portbaroakland.com/
Sunny Side Up Drag Brunch
Oakland Pride presents a new weekly dine and drag brunch starting Oct. 4, with host Ming Munro. $10-$100, Sundays 11am-2pm. 2044 Franklin St. https://www.eventbrite.com/e/sunny-side-up-brunch-with-oakland-pride-tickets-122904473589
Grace Cathedral
COMMUNITY, ACTIVITIES, CLASSES, PODCASTS, RADIO
AXIS Dance Classes
'If you have a body, you can dance' is the acclaimed Oakland disability-inclusive award-winning company. Take a Zoom class with company members JanpiStar, Bradford Chin, movement improvisation with Artistic Director Mark Brew, or a ballet with guest teacher Robert Dekkers, contemporary style with Bianca Cabrera. www.axisdance.org
GLAAD's new podcast focuses on the Black gay and queer male experience, hosted by Darnell Moore, producer of Being Seen and singer, actor and activist, Lawrence "Miss Lawrence" Washington. https://www.beingseenpodcast.com/
Castro Country Club's Virtual Gratitude Meetings and Check-in
Castro Country Club, the safe space for queers in recovery, will be holding virtual meetings on Sundays at 1 pm until they are able to reopen. The Zoom meeting room will open at 12:30 and the meeting room will remain open for at least 30 minutes thereafter for hanging out and fellowship. https://www.facebook.com/pg/CastroCountryClub/events/
Conservatory of Flowers
Online Zoom classes, and outside floral exhibits. https://conservatoryofflowers.org/exhibits-events
Daily Services at Grace Cathedral
Grace Cathedral, the welcoming and affirming Episcopalian congregation atop Nob Hill, will be holding online services weekday mornings at 9am and Sunday mornings at 11am. Grace is also hosting online Bible study and other events. www.gracecathedral.org
Fighting Racism
Join fundraising support and civil rights activism through various groups: The George Floyd Memorial Fund
(www.gofundme.com/georgefloyd, The Minnesota Freedom Fund (https://blacklivesmatter.com/|blacklivesmatter.com/), Campaign Zero (www.joincampaignzero.org).
Gay Men's Sketch/Hot Draw
Mark I. Chester hosts three gay men's sketch events a week, with sexy male models posing and works shared via Zoom. Ongoing events - Tue, 6:30pm: Gay Men's Sketch in exile. Fri, 6:30pm: Hot Draw! in exile. Sun, 2pm: Gay Men's Sketch in exile. instagram.com/gaymenssketch instagram.com/hotdrawsf
Fighting Back, a series which brings together community leaders, experts, historians and activists to explore lessons from the past that might be useful in formulating resistance efforts today. The 4127 18th St. museum opened Oct. 1, with advance reservations required. Online exhibits include Reigning Queens: the Lost Photos of Roz Joseph, curated by Joseph Plaster; Angela Davis: OUTspoken, Labor of Love about the first ten years of Pride events in San Francisco; Performance, Protest & Politics: The Art of Gilbert Baker, Pioneering Periodicals, 1940s-1950s, Picturing Kinship: Portraits of our Community by Lenore Chinn and 50 Years of Pride. https://www.glbthistory.org/fighting-back
GLIDE Services
Online services with the spiritual center of healing, faith, justice and community for everyone. https://www.glide.org/church/
Outdoor Dance Classes with Dance Mission
Revry's array of online films, activist chats and more, including Latinx LGBTQ personalities on the show The Q Agenda. https://revry.tv/
Job Services at LGBT Center
Learn new job search skills, work on your resume and more in online workshops. Oct. 9, 11am: Resume Workshop. https://www.sfcenter.org/program/employment_services/
The Jewish LGBTQ organization's online events include discussions, author talks and celebrations. https://www.keshetonline.org
LGBT Asylum Project Podcast
Creator and host of the LGBTQ Asylum podcast is 28-year-old Sayid Abdul, a queer Uighur refugee who was granted political asylum in 2018. Sayid speaks authentically and truthfully to the US asylum process, and he and his guests engage listeners in gripping, critical stories that bring urgency to a crisis affecting millions; available on Spotify and Apple's Podcast. https://www.lgbtasylumproject.org/
LGBTQ Connection
Online and in-person events for Napa and Sonoma community, including a voter registration party, candidate forums, Big Gay Trivia night, counseling and more. https://www.lgbtqconnection.org
Jeffrey Masters hosts a series of weekly podcasts with community notables; sponsored by The Advocate and GLAAD. https://podcasts.apple.com/us/podcast/lgbtq-a/id1151561226
Manny's Live
Community leaders, authors, activists and politicians join online discussions, voter registration campaigns and more. www.welcometomannys.com
National AIDS Memorial
The National AIDS Memorial launches a new web platform that features the entire AIDS Quilt's 48,000 panels online, with a new initiative to share stories from 40 years of the pandemic from then to now. July-ongoing. Opening is coordinated with the annual International AIDS Conference. https://aidsmemorial.org/
Out in the Bay
The long-running LGBT radio show, offline for a few years, has returned with new podcast episodes, and an extensive archive. Host Eric Jansen welcomes artists, community leaders and celebrities. https://www.outinthebay.org/
Outdoor Dance Classes
Stretch, spin, sing your body electric in outdoor, distanced playground-set SF/mission and West Oakland dance class fin Hip Hop, Salsa, Cuban and other styles. Also online classes. https://dancemissiontheater.org/
Queer Powers
Indie channel of chats radio shows, vintage zine archives and drag shows. https://www.queerpowers.com/
A Queer Serial
New series of LGBT-themed podcasts with voice actors such as Sam Pancake (Transparent, Gilmore Girls), @lgbt_history's Matthew Reimer, Radical Faeries' Will Roscoe and Joey Cain, and Chicago performers Salvio Gado and Jon Martinez, along with real recordings of pioneers including transgender entertainer Christine Jorgensen, civil rights activist Bayard Rustin, and lesbian journalist Edythe Eyde. https://www.mattachinepod.com/seasontwo
Queer Nightlife Talks
Blog and links to talks among Bay Area nightlife organizers and producers, with community input and opinions. Tune in, offer assistance or thoughts. Nov. 10, "How do you live without the party?" a public Zoom forum hosted by Angelique L. Burke and Ralph Brunei. sfqueernightlifefund.org
Queercore Podcast
The QueerCore Podcast, hosted by August Bernadicou, elucidates radical, LGBTQ activists' dark histories; a spotlight on the catalysts who fought in the front-lines, in the back-lines and in the trenches of civil rights. The QueerCore Podcast is an opportunity for them to tell their stories in their own words, including Phyllis Lyon, Don Kilhefner, Jack Fritscher, drag rocker Jayne County and Vanguard Editor Keith St. Clare. https://www.queercorepod.com/
Resisterhood
Website, documentary film and toolkit for women to restore their power, encourage voting and resistance. https://www.resisterhooddoc.com/
Respect Everywhere
GLSEN's LGBTQ+ talent and allies event showcases the incredible stories of students from around the country who are making a tremendous impact on behalf of the LGBTQ+ community in their own way, with videos directed by Kalen Allen, Matt Bomer, Rob Hardy, Joe Mantello, Antoni Porowski, and Octavia Spencer. Oct. 26. www.glsen.org/respecteverywhere
Roryography
For dance class fun right off the stage of some Oasis and Peaches Christ productions, Roryography is also now online. Sign up for classes at https://www.airbnb.com/experiences/1672375 or https://www.facebook.com/rory.davis.94.
Sexitude
D'Arcy Drollinger's fun retro-aerobic dance class returns, now in-person and outdoors at Dolores Park. http://www.sexitude.com/
Shabbat Services With Congregation Shaar Zahav
During the Covid-19 shutdown Congregation Shaar Zahav, San Francisco's synagogue for LGBTQ Jews and friends, will stream their Shabbat (Sabbath) services via Zoom. Friday nights at 7:30 pm and Saturday mornings at 10 A.M. https://shaarzahav.org/services/
Smuin Contemporary Ballet Classes
The local dance company streams notable works, and offers online dance classes in all levels of ballet, plus tap, hip hop, contemporary and dance for fitness, and Broadway Jazz classes with hunky Weston Krukow; through Oct. 31. $6-$20 each. www.smuinballet.org
MTV News brings together leading voices for "Sound On," a roundtable series igniting essential conversations through a BIPOC lens. Panelists dive into the topics of gender identity, white supremacy, allyship & more. https://www.youtube.com/watch?v=JPc5wy-iROM
Stay Fresh at Home
Fresh Meat Productions' new online weekly series of talks and classes to support your wellness, embodiment, breath and connection, with singing, dance and lessons in various topics. http://freshmeatproductions.org/
Taimi Talks
LGBTQ personalities (Jason Wimberly, Chris Mosier, Monique Heart and others) share guest-talks on various topics, with/on the Taimi app. (Apple Store or Google Play). https://taimi.com/
The T-House
Community gathering space for transmasculine and masculine of center individuals including trans men, nonbinary people, genderfluid people, butches, studs, two spirit people who identify with transmasculinity, sponsored by City College of San Francisco. Weekly online events. https://t-house.carrd.co/
Worship With Rev. Dr. Megan Rohrer
Rev. Dr. Megan Rohrer of Grace Lutheran Church, the first openly transgender pastor to be ordained in the Lutheran church, will be offering a variety of online services and events during the Covid-19 shutdown. For kids stuck at home there will be Kids Dance Parties on Zoom until May 1. Pastor Rohrer will also hold their Sunday services online via Facebook live at their Facebook page.
Zoom Baking Classes
Join a variety of notable chefs for fun tutorials on baking. Fridays 10am PT/1pm ET.
www.24sixlife.com/baking-zoom-room
Do you have an event to add? Email events@ebar.com
(Jim Gladstone contributed to theatre listings.)
Homing's In Oct. 29-Nov 5
When GAY was good: Pioneering weekly magazine covered community and arts scene
Blue Hearts beating: an interview with gay musician Bob Mould
Castro Merchants will seek to close parts of 18th Street for outdoor dining | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,147 |
Q: Floating nav buttons and margin 0 auto not displaying correctly I'm building a nav bar for my site. The site has 3 buttons. One on the left one on the center and one on the right. I have the left button floating left. The center button has margin: 0 auto to center it and the right button is floated right. There is a problem with the right button not sitting in the navigation bar. Its over flowing out of the bar. I have attached an image showing this.
A: I guess from your html markup that the right button is literally last, move it to be first, I.e before the left floated element, and it will sit as you expect
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,030 |
[](https://atmospherejs.com/ryanwu/meteor-notifyjs)
# Meteor Notify.js Package
## About
This is a [meteor](https://www.meteor.com/) package which contains lastest version of [Notify.js](https://github.com/jpillora/notifyjs)
## Installation
```
meteor add ryanwu:meteor-notifyjs
```
## Dependancy
* JQuery
## License
MIT: [http://ryanwu.mit-license.org](http://ryanwu.mit-license.org)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,436 |
Allium hintoniorum is a plant species native to the State of Nuevo León in northeastern Mexico.
The specific epithet hintoniorum is grammatically in a plural form, as the plant is named in honor of British-born botanist George Boole Hinton (1882-1943), his son Jaime, and his grandson George.
References
haemanthoides
Onions
Flora of Nuevo León
Plants described in 1994 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,078 |
Category Archives: Baal Haturim
5775, Baal Haturim, Genesis, Jacob, Parsha, The Times of Israel, Vayishlach
Idolatry Allergy
November 30, 2015 bentzis Leave a comment
Baal Haturim Genesis: Vayishlach
Whatever a man seeks, honors, or exalts more than God, this is the god of his idolatry. -William B. Ullathorne
I grew up with a variety of allergies. There were foods, substances and smells that would make me sick and specifically trigger respiratory difficulties. In my youth, during visits to a specific person's home, I would unexpectedly start to sneeze uncontrollably. I was informed afterwards that every time I would have a sneezing attack it coincided with someone smoking marijuana in a nearby room.
But perhaps the most surprising allergy of all was not to any particular molecule that made its way into my respiratory system. In the course of my teenage travels, I had opportunity to be exposed to real live examples of idol worship: people praying, chanting, bowing down and performing religious service in front of statues. Just the sight, the approach, just to be in the same physical space or structure as the good old-fashioned idolaters made me physically uncomfortable. At the time I was not yet aware of the prohibition by Jewish law of entering a Temple of idols. Nonetheless, my body, apparently of its own volition, reacted negatively to any encounter with these structures and activities, typically with feelings of nausea.
In the Torah we see a similar but broader national effect. Jacob's wayward brother Esau, together with his growing clan, leaves their ancestral land of Canaan. The Baal Haturim on Genesis 36:7 explains that because of the divine presence in Canaan, Esau could not stay there with his brother Jacob. He says that a similar effect occurred in the separation between the children of Israel and the Egyptians who needed to live in different areas. We cannot be in the same place as idolaters, and vice versa.
May we identify the idolatry in our lives and separate ourselves from it and from within us.
To my Tikvah Fund friends. It was thrilling to explore the Bible and discover excellence.
5775, 5776, Baal Haturim, Deuteronomy, Parsha, The Times of Israel, Vezot Habracha
Pre-Incarnated Unity
October 1, 2015 bentzis Leave a comment
The soul gives unity to what it looks at with love. -Thomas Carlyle
At what was perhaps the most transcendent moment in human history, God reveals Himself to the Jewish people at Mount Sinai, where the Ten Commandments are uttered and God gives the remainder of the Written and Oral Torah to Moses. Every single one of the Children of Israel who was alive at that time, shortly after the Exodus from Egypt, hears and senses God at prophetic levels.
There follows a question as to how this bond, this covenant that was formed at Sinai can continue through the long generations and millennia since that singular event. What connects, what unites the descendants of those who stood at Sinai with the ancestors who witnessed the barely filtered presence of God?
Amongst many answers, a popular one is that the soul of every Jew was at Sinai, even if he hadn't been born yet. Somehow, at this defining event for the Jewish people, every Jewish soul, alive as well as unborn, through all the generations, was present for the receipt of the Torah, for the establishment of the everlasting covenant with God.
The Baal Haturim on Deuteronomy 33:3 adds another facet to this well known explanation. He elaborates that not only was the soul of every future Jew in history present at Sinai, but that even the souls of future converts were present at the encounter. That the truth is that their souls were there as well, and heard and received the Torah. When they convert, they are merely reconnecting and reclaiming that spiritual heritage that was rightfully theirs from so long ago, where we all accepted the divine mission as one united people.
May our souls re-accept the Torah on a regular basis.
To the unifying Sukah of the Lefler family.
5775, 5776, Baal Haturim, Deuteronomy, Haazinu, Parsha, The Times of Israel
Forging the Eternal Inheritance
A man cannot leave a better legacy to the world than a well-educated family. -Thomas Scott
It is my sober duty to bury the dead and console the living. It gives one ample opportunities to ponder the legacy people leave behind. The family matriarch who lived to see great-grandchildren following in her footsteps of kindness. The grandfather who was a well-known joke-teller with grandchildren who continue with the same entertaining sense of humor. Or those who died before they lived to see successive generations but left behind memories of strength and joy. Or those who left behind money and property to then be squabbled over by the children.
From a multi-generation, decades long perspective, it is sobering to consider what a parent should inculcate in their children, what they should hope to see in their grandchildren, what are the greatest gifts one can bestow on their progeny that will have a positive and lasting impact on ones descendants and on the world.
The Baal Haturim on Deuteronomy 32:7 sets the goal of Torah scholarship as the pinnacle of what a parent can hope for. He explains that if a family merits to see three successive generations of Torah scholars, that gift, that accomplishment becomes an eternal inheritance, for the family and for the wider nation of Israel.
One of the great scholars of the past was once asked, how many years did it take him to accumulate the vast knowledge of Torah that was at his fingertips. He answered: "Five minutes." The questioner looked at the scholar in confusion. The scholar explained: "Whenever I was at a bus stop, whenever I was standing on line, whenever I had five minutes free, I would learn. Those five minutes added up."
If our fathers were not great scholars, it does not exempt us from striving, nor from setting an example for our children and grandchildren. It just takes five minutes.
Shabbat Shalom and Chag Sukot Sameach,
To the Chok Le'Israel. A book the provides really bite-size daily servings of the spectrum of Torah. Very highly recommended.
5775, 5776, Baal Haturim, Deuteronomy, Parsha, The Times of Israel, Vayelech
Afterlife Conversations
Baal Haturim Deuteromony: Vayelech
We talk about heaven being so far away. It is within speaking distance to those who belong there. Heaven is a prepared place for a prepared people. -Dwight L. Moody
For good reason, there is great uncertainty about the afterlife. I have yet to meet someone who has been there and back, who could give a personal account of what it was like. Some doubt its existence as there is no scientific proof. Others may believe in various Hollywood versions, inspired in part from classical poets like Dante and Milton.
The Jewish tradition has much to say about the next world. What's its purpose, who gets there and who doesn't, what do we do there, how long we're there for, and much more.
The Baal Haturim points out an interesting capability we will retain in the afterlife. In his commentary on Deuteronomy 31:1, he cites a conversation between Moses and the Patriarchs (Abraham, Isaac and Jacob), whereby Moses informs them that God has fulfilled his centuries-old promise to them, that their descendents, the Children of Israel, have finally inherited the Land of Israel. The Baal Haturim explains that the above account is the source that the dead talk to each other in the next world.
May our patriarchs and ancestors have good things to say about us, especially as we approach the Day of Judgement.
To the Chazan, Shaul Hochberger, on his divine conversations leading our prayers.
5775, Baal Haturim, Deuteronomy, Nitzavim, Parsha, The Times of Israel
Instant Global Cure
To reform a world, to reform a nation, no wise man will undertake; and all but foolish men know, that the only solid, though a far slower reformation, is what each begins and perfects on himself. -Thomas Carlyle
We are drowning in a sea of strife and pain. Wherever we look, whatever we read, we cannot avoid the disrespect, the insensitivity, the cruelty, and the mayhem of one human being to another.
One common reaction is to sigh a breath of resignation. I am too far away. I am too small. I am too insignificant to affect this fight. Another reaction is to complain. To curse the powers that be and all those who stand aside, as evils are committed uncontested.
There is a third, slower path, one that doesn't necessarily fix the suffering staring us in the face, not completely nor immediately. But it is a step. That path is called repentance.
The Baal Haturim on Deuteronomy 30:8 talks about repentance. We must first find what is wrong in us and fix it. If there is disrespect, insensitivity or cruelty in us, we must address it before we can presume to lecture others. However, the Baal Haturim states something surprising. He explains that if we can achieve complete repentance we shall see and experience immediate redemption.
May we strive for full repentance on a personal, familiar, communal and global basis.
Shabbat Shalom and Shana Tova,
To Rabbi Asher Weiss, a wise man.
5775, Baal Haturim, Deuteronomy, Ki Tavo, Parsha, The Times of Israel
Respect the Silence
Men are respectable only as they respect. -Ralph Waldo Emerson
I've noticed that the volume in a room is often in direct proportion to the comfort the people in the room feel. Parties can be unbearably loud. Cemeteries are appropriately quiet. However, there are multiple occasions where quiet is both expected and given. Official assemblies of all sorts will have their moments of quiet.
Synagogue prayer is a conundrum. On one hand, we want participants to be happy and comfortable. For many it is a great opportunity to catch up with friends, to relax and chat. However, we are also supposed to be there to pray to God.
Jewish law is unequivocal about talking during prayer – it is forbidden, besides being rude, insensitive, ego-centric and disturbing. Our sages went so far as to institute a special prayer bestowing great blessings upon those who are careful not to speak in the synagogue.
The Baal Haturim on Deuteronomy 26:19 takes things a step further. He explains that when one is focused on his prayer, it is as if he is constructing a crown made out of prayers which is then placed, as it were, upon God's head. However, subsequently, in some spiritual, mystical sense, that crown then returns to the prayerful person. However, those who instead of respecting the prayerful quiet choose instead to talk during the services, instead of receiving a divine crown, they will be punished. The punishment, states the Baal Haturim, is that they will receive thorns all over their body.
May chatter in the synagogue be diminished and may we be spared from any punishments.
To all the participants in the Maimonides Shabbaton. Thank you for a great Shabbaton and for quiet and meaningful prayers.
5775, Baal Haturim, Deuteronomy, Ki Tetze, Parsha, The Times of Israel
Ugly Language
The foolish and wicked practice of profane cursing and swearing is a vice so mean and low that every person of sense and character detests and despises it. -George Washington
One of the most defining characteristics of human beings are their ability to speak. And in a sense, the words we use define us, but not only the words that come out of our mouths, but also the words that we hear, the words that infuse our being. And in that area, like in so many others, we have free choice. We can choose both what to speak and what to hear. There are few situations where we are forced either to say something or to listen to something.
The Baal Haturim on Deuteronomy 23:14 warns about this matter and specifically about "nivul pe", literally "the disgustingness of the mouth". He states clearly and simply that if we suddenly find ourselves exposed to foul language, to things that are not appropriate to be spoken, we should simply stick our fingers in our ears or get up and leave.
We would not subject ourselves (or our children) to harmful fumes, substances or dangerous situations. The same is true regarding foul language. It is toxic, corrupting and disgusting. The fact that it is widely used and accepted by the masses does not make it any better. Do not accept it. Do not stand for it. Let it be known that you disapprove. You will be surprised by the positive reactions people will have to this small stand of principle. And if they don't get it, spend your time with more refined people.
To the incoming class of Midreshet and Yeshivat Torah V'Avodah. I expect you will hear many beautiful words from your teachers and administration.
5775, Baal Haturim, Deuteronomy, Parsha, Shoftim, The Times of Israel
Lions of Judah
August 21, 2015 bentzis 1 Comment
Baal Haturim Deuteronomy: Shoftim
There is something behind the throne greater than the King himself. –William Pitt, The Elder Chatham
The Bible itself as well as subsequent Rabbinic commentators have mixed feelings regarding a monarchy. On one hand it seems to be a command that the nation of Israel should have a king. On the other hand, it seems that a monarchy may only be established if the nation desires one. If the nation wants a king, then there are certain guidelines as to the qualifications of a king as well as what he can and cannot do.
Not much after the nation of Israel conquers the land of Israel, we have the death of Joshua and the loss of centralized leadership. That time period is known biblically as the era of the Judges when over the course of a few hundred years the nation of Israel descends into civil war, chaos and anarchy. However, with the subsequent establishment of the monarchy of Israel, we relatively quickly get to corruption, idolatry and oppression, and a few hundred years after that, destruction and exile. In the long term, the difference between not having a monarchy and having one seems to be the difference between social madness and organized social madness.
Nonetheless, the Bible gives a tremendous amount of respect and importance to the historical monarchy. The desire for a king and the need to follow one to the people's liking is the source of much drama in the biblical books of Samuel and Kings and leads to the schism between the southern tribes (Judah and Benjamin) that remained loyal to the House of David and the ten northern tribes that went through various non-Davidic rulers. The Baal Haturim on Deuteronomy 17:15 points out to us that the kings of Israel are meant to be descendants of the tribe of "Lions", the tribe of Judah (as David, Salomon and their descendants were – and would indicate the northern tribes were ostensibly in the wrong in following non-Judean rulers, despite God's command and repeated intervention in the election (and assassination) of the kings of the ten tribes).
The error of non-Judean kings was repeated again during the second Temple era after the Hasmonean Revolt, where the successful Maccabees took the helm of political leadership despite being a Cohanic non-Judean family. The initial victory turned to ashes generations later as the Hasmonean line became corrupt and ends with Herod, who while an impressive builder, was a greater enemy of the Jewish people.
May we merit leadership of noble traits and correct pedigree, be they kings or otherwise.
To the two lions who included Montevideo in their courageous Halachic Adventures: Dr. Ari Greenspan and Rabbi Ari Zivotofsky.
5775, Baal Haturim, Deuteronomy, Parsha, Re'eh, The Times of Israel
The Metaphysics of Charity
August 10, 2015 bentzis 2 Comments
Baal Haturim Deuteronomy: Reeh
There never was a person who did anything worth doing, who did not receive more than he gave. -Henry Ward Beecher
Being charitable is a Jewish value that is recorded already from the stories of our Patriarch Abraham. In the time of Moses it is codified as law, including the requirement to tithe. The Rabbis give further clarification as to the percentage and measurements of different agricultural donations that each farmer was expected to contribute.
The Baal Haturim on this week's Torah portion provides a number of pointers as to the metaphysical reality of charity. He states in Deuteronomy 12:19 that the act of giving charity leads directly to increased wealth. In Deuteronomy 15:8 he explains that if a person listens to and provides for the poor, God in turn will listen to and provide for the charitable person. The inverse is also true. If a person ignores the plea of the poor, God is likely to ignore the potentially charitable person.
Finally, the Baal Haturim on Deuteronomy 15:10 details that we should be careful to provide the solicitant what they need. He brings as an example the story of King David who when he was seeking refuge from the ire of King Saul escaped to the Cohanic city of Nov where they provided him with bread and a sword, two things he was in dire need of.
May we have the capacity and opportunity to be generous to those in need and may we see our generosity divinely and abundantly rewarded.
To Rachel and Shalom Berger on their abundant celebrations.
5775, Baal Haturim, Deuteronomy, Ekev, Parsha, The Times of Israel
Fashionable Resurrections
August 4, 2015 bentzis 1 Comment
Baal Haturim Deuteronomy: Ekev
If that vital spark that we find in a grain of wheat can pass unchanged through countless deaths and resurrections, will the spirit of man be unable to pass from this body to another? -William Jennings Bryan
It is a principle of Jewish faith that at some point in the future, the dead will come back to life. We have it listed as 13th of Maimonides 13 Principles of Faith: "I believe with complete faith that there will be a revival of the dead when it will rise up the will from the Creator, blessed be His Name."
This precept raises multiple questions:
In what body will we return?
Will we return old or young?
If we suffered the loss of a limb, will we return whole?
If you believe in reincarnation, which person will return?
And finally, will we return dressed or naked?
While I have faith that all of these questions will be taken care of satisfactorily, the Baal Haturim does provide in Deuteronomy 8:3 the answer to at least one of the questions. He states that the resurrected will return fully clothed. He gives the analogy to wheat. If a seed of wheat can be buried in the ground "naked", decompose, and return fully grown and "clothed" then so too, those destined to return from death will return fully clothed.
One less thing to worry about.
To all innocent victims of violence. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,946 |
Q: How can I switch between a row of 4 and 2 buttons? I have this code:
<Grid Grid.Row="2" x:Name="buttonGrid" Padding="10,0,10,0" HorizontalOptions="FillAndExpand" VerticalOptions="Center">
<Button x:Name="aButton" Style="{StaticResource pointButton}" Text="Don't Know" />
<Button x:Name="bButton" Style="{StaticResource pointButton}" Text="Very Hard" />
<Button x:Name="cButton" Style="{StaticResource pointButton}" Text="Hard" />
<Button x:Name="dButton" Style="{StaticResource pointButton}" Text="Easy" />
<Button x:Name="nButton" Style="{StaticResource pointButton}" Text="Don't Know" />
<Button x:Name="yButton" Style="{StaticResource pointButton}" Text="Easy" />
</Grid>
I would like to have either the first four or the last two buttons appear at the bottom of the screen in a row.
Is it possible to have more than one Grid.Row="2" and switch to either one or the other with an IsVisible in the back end C#? I think this would do what I need but not sure if it's a good way to do this?
I'd appreciate some suggestions as right now I am just setting the IsVisible for each of the buttons and it doesn't display them in a row as expected.
A: You definitely can have two or more grids/containers be assigned to a given row. Using isVisible to show/hide those containers that hold groups of buttons should work just fine.
A: What you want is something like this
<Grid>
<Grid.RowDefinitions>
<RowDefinition Height="9*"/>
<RowDefinition Height="1*"/>
</Grid.RowDefinitions>
<StackLayout Orientation="Horizontal" HorizontalOptions="CenterAndExpand" Grid.Row="1" IsVisible="{Binding FirstButtons}">
<Button x:Name="button1" Text="button1" Command="{Binding ChangeButtonsCommand}"/>
<Button x:Name="button2" Text="button2" Command="{Binding ChangeButtonsCommand}"/>
<Button x:Name="button3" Text="button3" Command="{Binding ChangeButtonsCommand}"/>
<Button x:Name="button4" Text="button4" Command="{Binding ChangeButtonsCommand}"/>
</StackLayout>
<StackLayout Orientation="Horizontal" HorizontalOptions="CenterAndExpand" Grid.Row="1" IsVisible="{Binding SecondButtons}">
<Button x:Name="buttonA" Text="buttonA" Command="{Binding ChangeButtonsCommand}"/>
<Button x:Name="buttonB" Text="buttonB" Command="{Binding ChangeButtonsCommand}"/>
</StackLayout>
</Grid>
With ViewModel something like this
public class MainViewModel : ViewModelBase
{
public ICommand ChangeButtonsCommand { get; private set; }
private bool firstButtons = true;
public bool FirstButtons
{
get { return firstButtons; }
set
{
firstButtons = value;
OnPropertyChanged();
}
}
private bool secondButtons = false;
public bool SecondButtons
{
get { return secondButtons; }
set
{
secondButtons = value;
OnPropertyChanged();
}
}
public MainViewModel()
{
ChangeButtonsCommand = new Command(() =>
{
if (FirstButtons)
{
FirstButtons = false;
SecondButtons = true;
}
else
{
FirstButtons = true;
SecondButtons = false;
}
});
}
ViewModelBase is here https://pastebin.com/aLMXRN50
This gives you a row of 4 buttons, when you press any of them, it switches to only 2 buttons. There are many ways to achieve this, but I think that is what you are after. Note that the Grid.Row property is set in the child elements inside the Grid, not on the Grid itself.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,915 |
This is a list of all Grand Prix tournaments, which have been held for the Magic: The Gathering game. By the end of 2018, 654 Grand Prix events have been held. From the beginning of 2019, Grand Prix events became a part of a larger event, named MagicFests.
List of Grand Prix
Key
Grand Prix by country
The totals for each of the three columns will be different from each other, for several reasons:
Team Grand Prix have multiple winners, possibly from different countries.
The same individual may win multiple Grand Prix.
Future Grand Prix will already have been scheduled at a host city but not yet have a winner.
See also
Grand Prix
The DCI
References | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,410 |
The LAC itself offers free legal advice to members of the general public. Each session requires five volunteers: two groups of two volunteers each supervised by a qualified solicitor. The final volunteer works on reception, welcoming clients and getting them to fill in a short form describing their legal issue.
The process is quite simple for the client with consultations taking a maximum of one hour. The client is first interviewed, then their enquiry is researched before the students provide feedback and information back to the client. All this is done with the help of the qualified solicitor if needed.
At the Legal Advice Clinic I had the opportunity to interview clients and give them legal advice, which is a great insight into what I will be doing once I start working in a law firm.
Approximately 60% of the time the LAC is able to solve the clients' queries instantly. Should the client require more specialist advice, they run evening sessions on Thursdays. The resolution rate rises to approximately 90% after meeting with the client a second time.
As volunteer places are limited, getting involved can be a very competitive process – with only one in three student applicants being successful. Applicants must complete a detailed application form and then, if they are successful, attend an interview. Successful candidates are then given dates for two mandatory training sessions. Watch our students discuss the LAC in the video below or read a range of case studies showing how our students explore the law.
For LLB (Hons) Law student Birgit Eisman, volunteering at the LSBU Legal Advice Clinic has been the highlight of her studies at LSBU so far. "It is an experience that I believe will stay with me throughout my career. Being able to provide advice in-person to genuine clients with genuine needs, whilst still studying, has been at times exhilarating, at times nerve-wracking, and always rewarding."
Not only has the experience been unique for Birgit, but she has been told by several prospective employers that it helps her CV stand out from the pack. With training contracts exceptionally competitive for law graduates, securing work experience before graduation can be vital – and the LAC helps students do just that.
For Akim Tejan-Cole, after hearing plenty of positives from other students who were volunteering at the LAC, he did not hesitate to apply when the opportunity arose.
"Working at the clinic has been a great experience. It gives me the chance to put the legal skills I've learned into practice whilst giving something back to the community I live in. Most importantly, it provides me with the legal experience I need to be better prepared for my future legal career. I really believe that volunteering at the Legal Advice Clinic sets me apart from my peers."
Fatima Bello is currently in the second year of her Law degree at LSBU. She heard about the opportunity to become a volunteer at the LAC late in her first year after Alan Russell, one of the LAC Directors, encouraged her to apply. It turned out to be a great decision.
No other legal work experience I have done has given me this level of responsibility and client contact.
"This experience is so rewarding and beneficial. We are providing a great service to our local community whilst learning a great deal ourselves. I have been taught so many new skills from the experienced solicitors that supervise us - the training we receive is of an exceptionally high standard. No other legal work experience I have done has given me this level of responsibility and client contact. Working at the clinic has really boosted my confidence. It is helping steer me towards becoming the type of lawyer I would like to be. It is truly an honour to be a student at LSBU and a volunteer at the Legal Advice Clinic."
LSBU's Legal Advice Centre (LAC) is run by the University's Division of Law, part of our School of Law and Social Sciences. Read more about how to access free legal advice as a member of the community. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,706 |
import * as React from 'react'
import CopyToClipboard from 'react-copy-to-clipboard'
import { WithStyles, withStyles, Typography } from '@material-ui/core'
import AppSettings from '../../appSettings'
import EditorLayout from '../../components/EditorLayout'
import { Menu } from '../../components'
import { HtmlHelper, Aux } from '../../utils'
import { SettingService } from '../../services'
import { IJwtReadOnlyApiKeys, IFetchResult, IMenuItem } from '../../interfaces'
import { CopyIcon, CheckIcon, RecycleIcon } from '../../components/SvgIcons'
import CmsButton from '../../components/CmsButton'
import { crudEdit } from '../../jss/main'
export interface IJwtReadOnlyApiKeysSettingsProps extends WithStyles<typeof crudEdit> {
edit: boolean
}
export interface IJwtReadOnlyApiKeysSettingsState {
jwtReadOnlyApiKeys?: IJwtReadOnlyApiKeys,
loading: boolean,
primaryKeyCopied: boolean,
secondaryKeyCopied: boolean,
menuItem?: number
menuItems: { [id: number]: IMenuItem }
}
class JwtReadOnlyApiKeysSettings extends React.Component<IJwtReadOnlyApiKeysSettingsProps, IJwtReadOnlyApiKeysSettingsState> {
form: any
defaultMenuState: { [id: number]: IMenuItem } = {
[0]: { menuItem: 0, text: 'Jwt Read Only API Keys', selected: true, valid: true }
}
defaultState: IJwtReadOnlyApiKeysSettingsState = {
loading: true,
primaryKeyCopied: false,
secondaryKeyCopied: false,
menuItems: this.defaultMenuState
}
constructor (props: IJwtReadOnlyApiKeysSettingsProps) {
super(props)
this.state = this.defaultState
this.loadJwtReadOnlyApiKeySettings()
this.onPrimaryKeyCopy = this.onPrimaryKeyCopy.bind(this)
this.onSecondaryKeyCopy = this.onSecondaryKeyCopy.bind(this)
this.onRegeneratePrimaryKey = this.onRegeneratePrimaryKey.bind(this)
this.onRegenerateSecondaryKey = this.onRegenerateSecondaryKey.bind(this)
this.handleRegenerateKeyResult = this.handleRegenerateKeyResult.bind(this)
}
handleRegenerateKeyResult = (message: string, response: IFetchResult<IJwtReadOnlyApiKeys>) => {
if (response.ok) {
this.setState({
...this.state,
jwtReadOnlyApiKeys: response.data,
loading: false
})
AppSettings.success(message, { beep: false })
} else {
AppSettings.error(<div>Something went wrong regenerating your Jwt Token Key!</div>, AppSettings.AlertSettings)
}
}
public render () {
return <EditorLayout
menu={<Menu menuItems={this.state.menuItems} purpose='Edit' />}
content={this.getForm()}
horizontalButtons={null}
title={<span className='page-title pl-3'>JWT ReadOnly API Key Settings</span>}
hasEdited={false}
loading={this.state.loading}
saving={this.state.loading}
goBackUrl='/administration/settings'
/>
}
private loadJwtReadOnlyApiKeySettings () {
SettingService.getJwtReadOnlyApiKeys()
.then((response: IFetchResult<IJwtReadOnlyApiKeys>) => {
this.setState({
...this.state,
jwtReadOnlyApiKeys: response.data,
loading: false
})
})
.catch(error => {
throw error
})
}
private onRegeneratePrimaryKey () {
SettingService.postJwtReadOnlyPrimaryApiKey()
.then(response => this.handleRegenerateKeyResult('Successfully regenerated ReadOnly Primary Key, restart the web app for the changes to take effect.', response))
.catch(error => {
throw error
})
}
private onRegenerateSecondaryKey () {
SettingService.postJwtReadOnlySecondaryApiKey()
.then(response => this.handleRegenerateKeyResult('Successfully regenerated ReadOnly Primary Key, restart the web app for the changes to take effect.', response))
.catch(error => {
throw error
})
}
private onPrimaryKeyCopy = () => {
this.setState({
...this.state,
primaryKeyCopied: true
}, () => setTimeout(() => {
this.setState({
...this.state,
primaryKeyCopied: false
})
}, 5000))
}
private onSecondaryKeyCopy = () => {
this.setState({
...this.state,
secondaryKeyCopied: true
}, () => setTimeout(() => {
this.setState({
...this.state,
secondaryKeyCopied: false
})
}, 5000))
}
private getKeyCopyToClipboard = (
copied: boolean,
key: string,
onCopy: any) => {
if (this.state.loading) {
return null
}
const { classes } = this.props
let copyIcon = <span key='icon'>{CopyIcon(classes.buttonSvgMargin)}</span>
let copyText = <span key='text'>Copy Key</span>
if (copied) {
copyIcon = <span key='icon'>{CheckIcon(classes.buttonSvgMargin)}</span>
copyText = <span key='text'>Copied Key</span>
}
return <CopyToClipboard text={key} onCopy={onCopy.bind(this)}>
<CmsButton size='small' color='secondary' components={[copyIcon, copyText]} variant='text' />
</CopyToClipboard>
}
private getForm () {
const {
loading,
jwtReadOnlyApiKeys,
primaryKeyCopied,
secondaryKeyCopied
} = this.state
const {
classes
} = this.props
if (!jwtReadOnlyApiKeys || loading) {
return null
}
return <Aux>
<div>
<Typography variant='title'>Primary Key</Typography>
<br/>
<Typography>{jwtReadOnlyApiKeys.primaryKey}</Typography>
</div>
<div>
{this.getKeyCopyToClipboard(primaryKeyCopied, jwtReadOnlyApiKeys.primaryKey, this.onPrimaryKeyCopy)}
{HtmlHelper.renderCmsButton(this.onRegeneratePrimaryKey, RecycleIcon(classes.buttonSvgMargin), 'Regenerate', true)}
<br/>
<br/>
</div>
<div>
<Typography variant='title'>Secondary Key</Typography>
<br/>
<Typography>{jwtReadOnlyApiKeys.secondaryKey}</Typography>
</div>
<div>
{this.getKeyCopyToClipboard(secondaryKeyCopied, jwtReadOnlyApiKeys.secondaryKey, this.onSecondaryKeyCopy)}
{HtmlHelper.renderCmsButton(this.onRegenerateSecondaryKey, RecycleIcon(classes.buttonSvgMargin), 'Regenerate', true)}
</div>
</Aux>
}
}
export default withStyles(crudEdit)(JwtReadOnlyApiKeysSettings)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,604 |
Daisy Dares You
BBC Sound of 2010 Insight Part 13 – Daisy Dares You
Submitted by Mark / 11 years ago / New Singles, New Songs
Daisy Coburn, who goes by the name of Daisy Dares You (it seems she was a fan of ZZZap!), hasn't got a record deal yet, but the scramble to acquire her autograph is as fierce as has been seen in recent times. "We've never experienced anything this quick," said a spokeswoman from her management company,… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,253 |
{"url":"http:\/\/tex.stackexchange.com\/questions\/43347\/getting-includeonly-functionality-with-import-package","text":"# Getting \\includeonly functionality with import package\n\nI had problems compiling with the `\\includeonly` `\\include` system since I had images in the document form different subdirectories. For example the main file would be at `\/Main\/MainFile.tex` and the included file at `\/Main\/Chapter1\/Chapter1.tex` and one of the images would be at `\/Main\/Chapter1\/Images\/Image1.jpg` which was actually loaded through the `Chapter1` file as `\\includegraphics{.\/Images\/Image1}`.\n\nI had the problem fixed with the `import` package using `\\subimport{}{}`, but I don't have the `include` functionality of different `.aux` files which keeps the pagination and cross references intact.\n\nHow do I get the best of both?\n\n-\n\n## 1 Answer\n\nYou can tell LaTeX where to look for included pictures, using the `\\graphicspath` setting. For example,\n\n``````\\graphicspath{{\/data\/experiments\/plots\/}{\/data\/simulations\/}}\n``````\n\nwill prompt LaTeX to look for pictures relative to both these folders. You can then include e.g. `\/data\/experiments\/plots\/first\/straightline.pdf` as `first\/straightline`, and `\/data\/simulations\/rollthedice.pdf` just as `rollthedice`.\n\nIn this way, you can avoid typing out very long absolute paths every time, and yet you don't have to change anything if you move parts of your document from one directory to another.\n\n-","date":"2015-12-01 18:34:59","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\": 0.9011356234550476, \"perplexity\": 1771.8656242197026}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-48\/segments\/1448398468971.92\/warc\/CC-MAIN-20151124205428-00111-ip-10-71-132-137.ec2.internal.warc.gz\"}"} | null | null |
**Contents**
_Title Page_
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
Chapter 29
Chapter 30
_Burdens of Empire By C.J. Ryan_
_Preview Of Burdens of Empire_
_About the Author_
_Also by C.J. Ryan_
_Copyright Page_
THE LIGHT OF THE TWO SUNS BEAT DOWN ON the baking streets of Cartago like a curse. One of them was small, blue, and hot; the other was fat, red, and less hot. They were separated in the sky by about the width of a fist held at arm's length, and Cartago whirled endlessly about a point somewhere between the two blazing orbs. In the streets they cast twin shadows, which angled away from one another slightly, as if reluctant to admit they knew each other.
Gloria VanDeen followed her host and his entourage through the swarming streets, enjoying the color and bustle and the singsong entreaties of the bazaar merchants. Exotic, fantailed phoenixbirds could be hers for only ten crowns, or perhaps even seven. Silken scarves and golden bangles and cotton caftans of the highest quality would only enhance her astonishing beauty, and at a price so low she would feel guilty for taking such advantage of the poor but honest merchants of Cartago; but the honor of being permitted to offer garments and jewelry to grace the famous form of Gloria VanDeen would more than offset the financial loss. The most beautiful woman in all the Empire surely deserved no less than the finest craftsmanship and artistry in the entire Sector—nay, the entire Quadrant!
"I would buy you something as a souvenir," Praetor Ulmani said to her, "but that would surely cause a riot. The merchant whose wares you wore would become insufferable, and his competition, unable to bear their shame, would doubtless kill him."
"Then I suppose it's best that I wear nothing," Gloria said with a smile. "I wouldn't want to be responsible for a tragedy."
"Spirit forbid it!" Ulmani grinned at her. Since nothing—or very nearly nothing—was precisely what Gloria was wearing, the comment seemed apt. Her soft, flowing, nearly transparent garment rested lightly on her shoulders and descended in narrow vees of sheer white fabric, front and rear, shielding her from the suns but not from the hungry eyes that surrounded her.
Honestly, there were times when being Gloria VanDeen was just so damn much fun that it should have been illegal. Gloria smiled and waved happily to the throng in the streets, then let Ulmani take her elbow and usher her along.
The people of Cartago, like their Praetor, had many reasons to be grateful to Gloria VanDeen, and Gloria was aware that her beauty and sexual presence were prominent among them. Without those attributes, it was doubtful that she would have been able to fulfill the mission that had brought her here.
On this, the thirteenth day of January in the year 3218, Standard Calendar—just three days after her twenty-fifth birthday—Gloria VanDeen was the most famous, popular, and (quite possibly) important woman in an empire that spanned a sphere of space two thousand light-years in diameter and was home to some 3 trillion sentient beings. The fact that she was the former wife of the man who was now Emperor Charles V was of some importance, as was her position as head of the Office of Strategic Intervention—the newest action arm of the Department of Extraterrestrial Affairs, the sprawling bureaucracy that administered the Terran Empire. Alone, either one of those facts would have made her a woman of some consequence; together, they made her a potent force. But it was her beauty, brilliance, and courage that had won the hearts of the masses and made her, as was often said, the Sweetheart of the Empire.
At moments like this, Gloria was outrageously happy to be exactly who and what she was. She had just brought her mission on Cartago to a triumphant conclusion. And for a change, the intervention had been brief, bloodless, and relatively simple.
Cartago was a thirsty world, where a small population of some 5 million lived on the margins of a globe-girdling desert. Just 194 light-years from Earth, the planet had been settled some eight hundred years earlier in pre-Imperial days. In later years, it probably never would have been colonized at all, but in that first era of interstellar expansion, Earthmen were not very particular in their choice of new worlds. If a planet had oxygen in its atmosphere and a mean surface temperature somewhere between the freezing and boiling points of water, it was a candidate for settlement. Cartago qualified, if only just.
Over the centuries, the slowly growing population had splintered into what amounted to tribal groups, although the ethnic, cultural, and religious orientations of the three main tribes differed little. The only differences of real consequence concerned water and access to it. The Mountain Tribe controlled the flow of the precious liquid that trickled down to the dusty plains and the lands of the Eastern and Western Tribes. Accordingly, the Mountain Tribe had always selected the Praetor who headed the planetary government, such as it was. The coming of the Empire had remarkably little impact on Cartago, and generations of Imperial Governors and Dexta bureaucrats were content to let the natives work things out for themselves.
In recent years, however, the elders of the Eastern and Western Tribes had—for Spirit knew what bizarre reasons—taken to sending their brightest sons and daughters back to Earth to be educated in the law. Having nothing better to do when they returned to their homeworld, the young barristers began suing the other tribes over the only thing that mattered on their world—water. Thus, there had been angry protests, boycotts, insults, threats, and finally, the sequestration of water that had once flowed freely from the slopes of the central mountains. The Imperial Governor had been unable to persuade anyone to see the light of reason, and the tiny Dexta establishment on the planet had thrown up its arms in despair.
So Norman Mingus had sent Gloria to Cartago.
A year and a half earlier, before the creation of the OSI, Gloria—then a Level XIII Dexta bureaucrat—had used her wit and wiles to attract a thousand volunteers on the planet Pecos to help her avert a genocide on the backwater world of Mynjhino. Dexta Secretary Norman Mingus, taking note of her success, had recognized the extraordinary power she possessed and sought to employ it on a regular basis by appointing her to lead the new Office of Strategic Intervention. He had planned to use Gloria as his chessboard queen, dispatching her hither and yon throughout the Empire to hot spots where the existing bureaucratic machinery was failing to respond adequately to the challenges of the moment. In the year since the formation of the OSI, Gloria had only burnished her already gleaming reputation as the sexiest and most desirable woman in the Empire—and, in the process, had achieved some remarkable results for the benefit of Dexta and the Empire it served. She was wildly popular in every corner of the Empire—except, perhaps, for the Imperial Household itself, where Charles had yet to reconcile himself to the new role being played by the woman who had walked out on him nearly seven years earlier.
Cartago was the fifth intervention for Gloria and OSI, and by far, the easiest. Gloria hadn't even bothered to take along any staff; Cartago was just two days away from Earth via Flyer, and so help was relatively close at hand should any prove necessary.
None did. In less than two weeks on the planet, she had met with a handful of young lawyers, a clutch of tribal elders, and Praetor Ulmani, adroitly resolving their conflicts through a combination of judicious bribes and personal persuasion. The Dexta Comptroller would probably grumble about the bribes, small though they were, but Gloria had thoroughly enjoyed her sojourn on Cartago.
In fact, resolving the conflicts on Cartago had been so easy that Gloria seriously wondered if the whole mess had been cooked up by the young lawyers simply to lure her to their world. Probably not, she conceded; on a desert world, no one played games with water. Still, her fame and reputation had reached the point that she had to beware the Heisenberg Effect; her mere presence was enough to alter the terms of any equation.
At the moment, her presence in the streets of Cartago was altering the normal routine of the bazaar at searing midday, when energy levels and activity normally reached low ebb. Praetor Ulmani kept his grip on Gloria's elbow while his entourage of aides and security men plowed a path through the increasingly excited crowds. They could have taken an air-conditioned limo skimmer directly to the restaurant, but Ulmani had calculated that it could not hurt his popularity to be seen escorting Gloria through the teeming streets of his city. And Gloria didn't really mind being paraded as if she were one of Ulmani's hunting trophies; she was willing to let him get whatever political mileage he could out of the affair. He was a nice enough man and probably as good a leader as the planet needed. She was content to let him have his moment; all that really mattered to Gloria was that she had solved Cartago's problems and would soon be returning to Earth.
The outdoor restaurant was just ahead. Security men were already clearing a path while the maître d' stood before a large table in eager, fawning anticipation. Gloria impulsively decided to give everyone—not least of all Ulmani—one last reason to cheer. She turned to face Ulmani, pulled him close, and gave him an incendiary kiss. The cheers were deafening. Gloria released her hold on the astonished Praetor and grinned at him. "Your people adore you," she said.
"After this," Ulmani replied, "they may start worshipping me!"
They made their way through the wrought-iron gateway that delineated the restaurant, greeted the proprietor, and moved to the reserved table. One of Ulmani's aides pulled out a chair for her, and Gloria had just bent to be seated when the blue-green bolt of plasma crackled through the air just above her. The characteristic plasma thunderclap followed an instant later, as air rushed in to fill the ionized trail of the discharge. And just after that, Gloria heard the pained, startled moan from the aide standing behind her.
Before she could fully register what had happened, three security men were firing back, their plasma beams converging on a head and shoulders just visible on the rooftop across the street. The head and shoulders vanished in green fire, and the triple thunderclap echoed through the suddenly silent streets. As quickly as that, it was over.
Gloria was suddenly aware of the sharp smell of burned hair.
She looked around and saw Ulmani's aide slumping backwards against the white stucco wall of the building. Flames flickered along the torn sleeve of his garment and he stared in openmouthed wonder at the inch-in-diameter trench of blackened meat that now grooved his upper arm. Behind him, there was a smoldering hole in the wall.
Comprehension dawned, and Gloria slowly reached upward and ran her fingers through the singed tunnel that had scorched through her thick blond mane, just above her scalp.
_I've been shot in the hair,_ she thought. _How very odd._
GLORIA STOOD NAKED ON THE ALTAR BEFORE fifty thousand entranced onlookers and an Empire-wide vid audience that would eventually number a trillion or more. As it was a Visitation Day, everyone else in the cavernous Church of the Divine Spirit was naked, too, including Archbishop Nesselrode, who stood next to her as he droned on and on and on. What was there about religion, she wondered, that turned so many people into compulsive blabbermouths? Instead of standing in silent awe before their conception of the infinite, why did people like the Archbishop feel the need to heap platitude upon platitude until the stack reached all the way to the heavens above?
The Spirit Herself, at least, had been relatively concise. Exactly eleven hundred and one years ago, to the very hour—at this very spot—she had made her first Visitation, in the middle of what was then a soccer stadium. She had spoken for just ten minutes, and, in her six subsequent Visitations, never more than fifteen. It made for a rather short gospel. In fact, the Book of the Spirit was a remarkably slim volume for a document that had transformed human history.
After centuries of bloody conflicts, many of them inspired by religion, the Spirit had launched a new religion in the year 2117 that was now embraced by fully 70 percent of the Empire's population. It had been eleven centuries since the last religious war among humans, and no Spiritist had ever sponsored a pogrom, crusade, inquisition, jihad, or witch-hunt. The Spirit had made no miraculous claims nor offered magical spells and incantations; she demanded no extraordinary sacrifices and condemned only intolerance, greed, and stupidity. She threatened no hell and promised no heaven but the ones humans could make for themselves. Her gospel was gentle, worldly, even sensuous.
The Universal Church of the Spirit had grown in the wake of her Visitations and was now centered here, at the Mother Church in Rio de Janeiro. The church itself was one of the largest buildings on Earth, its main spire soaring nearly a kilometer above the rotating, circular altar where Gloria now stood, surrounded by legions of the faithful. She had come here today, in spite of many misgivings, to be honored for her life and her work—for her personification of the ideals and values espoused by the Spirit Herself.
It was ridiculous, of course. However, it also seemed unavoidable.
She had been putting off invitations from the UCS ever since her return from her mission to Sylvania, nearly six months ago. Events on that far-off world had been seized upon by devout Spiritists as living proof of the reality of the Spirit, and Gloria's role in those events had inspired their admiration. Gloria was all too aware of the underlying reality of those events but was hardly in a position to reveal the truth to the rest of the Empire. So here she was at last, very much against her will, being honored by people who would probably lynch her if only they knew the truth.
Some people even thought that Gloria resembled the Spirit. There was, perhaps, some truth in that assessment, if one ignored the fact that the Spirit's hair was raven while Gloria's was golden, and that Gloria was five-and-a-half feet tall to the Spirit's forty. But the Spirit's face, like Gloria's, was vaguely panracial in appearance, and her immense, nude body was every bit as attractive as Gloria's, although the Spirit's breasts were proportionately somewhat larger and her hips a bit broader.
But Gloria was real. The Spirit, according to the Cynics, was nothing more substantial than a clever holographic projection arranged by a band of maverick scientists, and her gospel nothing but a collection of slick catchphrases composed by advertising copywriters.
Gloria had always wanted to believe in the Spirit. Sometimes, she almost did. But after Sylvania, she now doubted that she ever really could. Sylvania had taught her how easy it was to create a fake deity and how eager people were to believe in one. Given the role that she had played in the fraud that was the Voice, Gloria could only assume that the Spirit had been a similar fraud.
Ironically, however, the Spiritists' enthusiastic acceptance of the Voice had probably forestalled an investigation that would undoubtedly have concluded with Gloria's spending the rest of her life on a prison world for high treason. It was therefore expedient for Gloria, finally, to accept the honors the Church wanted to bestow upon her. And in the long run, given her ambitions for her future at Dexta, it couldn't hurt to have close ties with the Spiritists. But still, it bothered her to be implicated so deeply in promoting the fraud of the Voice—and, presumably, the fraud of the Spirit.
But if the Spirit truly was a sham, it had been an incredibly effective and beneficial one—as had the Voice. The Voice had helped save a planet from devastation, while the Spirit had probably kept the human race from destroying itself over a thousand years ago. Gloria found that she could not condemn a benevolent lie, any more than she could wholeheartedly embrace a destructive truth. People, in any case, would insist on believing whatever they wanted to believe. Today, they wanted to believe that Gloria VanDeen was an Avatar of the Spirit.
The Avatars of the Spirit (Gloria was about to become the 129th) were flesh-and-blood exemplars of the words and wisdom of the Spirit—in effect, they were saints for a religion that had no saints. Gloria didn't much care for the notion of becoming an Avatar of the Spirit, but the ceremony today would make her status official and eternal, as far as the Church was concerned.
During each of her Visitations, the Spirit had spoken of one of the Seven Seeds of Wisdom: Love, Compassion, Tolerance, Generosity, Knowledge, Joy, and Peace. Over the centuries, each of the Seeds had formed the core value of one of the Seven Septs of Spiritism. Members of the Church naturally gravitated to one or another of the Septs; doctors tended to embrace the Sept of Compassion, scholars belonged to the Sept of Knowledge, and so on. The Septs were not exactly separate denominations; they coexisted comfortably and compatibly under the broad tenets of Spiritist theology. The Sept of Joy was sponsoring Gloria's installation as an Avatar.
"Do not deny yourself Joy," the Spirit had counseled, "for it is a gift unto humankind and the wellspring of happiness." Theologians had generally taken "joy" to mean "sex," although the Spirit had never used precisely that word. The very fact that the Spirit had been nude argued that the human body was nothing to be ashamed of, and her warm endorsement of "joy" was taken as a sign that there was nothing sinful about sex. Indeed, some of the early Avatars of the Sept of Joy were courtesans, and one was an empress who earned a reputation rivaling that of Messalina. Gloria didn't think that she quite lived up to the standard set by her predecessors, and wasn't sure that she wanted to, but she could hardly deny that her sex life was as public and spectacular as anything the early thirty-third century had to offer.
Gloria had grown up at Court, under the reign of old Darius IV, when public nudity and sex were not only accepted but expected. Things had become more restrictive during the brief reign of grim, dour Gregory III; but now, under Charles, overripe decadence was once again the order of the day. That was fine with Gloria, who appreciated sex as much as anyone—more, in fact, since she possessed certain genetic enhancements that dramatically magnified her ability to give and receive pleasure. But she wasn't sure how she felt about her own sexuality being officially recognized and all but sanctified by the Church. Somehow, she felt, her volcanic orgasms ought to be a personal matter, if not necessarily a private one.
Archbishop Nesselrode reached his peroration, at long last, and Gloria leaned forward slightly as the Archbishop placed a gold chain necklace over her head. Depending from it was a ruby-red mustard seed, since the ruby was the designated gemstone of the Sept of Joy. With this symbolic act, Gloria was now confirmed as an Avatar of the Spirit.
Fifty thousand voices roared their approval. But Gloria could only stand there and smile wanly as the altar slowly rotated to let everyone view the Church's newest Avatar.
_Holy shit,_ she thought.
FOLLOWING THE CEREMONY, GLORIA JOINED with Church officials and the media at a reception. She sipped wine and nibbled canapés while a procession of naked men and women shook her hand, offered their congratulations, and wished her continued Joy. Many, it was clear, wanted to share that Joy with her; a few seemed ready and willing to do it then and there. Gloria graciously declined.
She wondered if it was going to be this way from now on. She had mixed feelings about the public reputation she had acquired in the past year or so. At times she adored it, but there were others when she wished she could go back to being plain old Gloria VanDeen, anonymous Avatar of Nothing and Nobody's Sweetheart.
But no, that was ridiculous too. She was, after all, the former wife of the man who was now Emperor, and royals didn't wed anonymous nobodies. Gloria was a member of a wealthy and prominent family and her beauty had attracted boys and men ever since she first blossomed. She had a face that could probably launch about 990 ships, if not a thousand, and a body that could launch a thousand more. With a genetic heritage from six continents, she boasted a flowing blond mane, flawless skin the shade of coffee with a little cream, and arresting, exotically angled eyes the color of polished turquoise. Her lips were perhaps a little thin for perfection, but they curved upward slightly at the sides, in a hint of a permanent, bemused smile. Her body was slim and athletic, her breasts not large but firm, globular, and tipped by erect, cylindrical nipples. Her navel was artistically whorled; thanks to her genetic enhancements, it was—like her nipples—as erotically sensitive as the primary sexual features of any normal woman. She could achieve orgasms in mere seconds, if she chose, or drag them out endlessly until sheer exhaustion finally brought them to a halt. Sex with Gloria VanDeen was an experience no one ever forgot.
She was inescapably special, and knew it. But still...an Avatar of Joy?
Gloria sighed, shook her head slightly to clear away the reveries, and tried to focus on the next person who wanted to meet her. This one, anomalously, was wearing clothing, Visitation Day notwithstanding. And not just any clothing—he was decked out in the Imperial livery of the House of Hazar.
The young man leaned close and whispered in her ear. "Pardon the intrusion, Ms. VanDeen, but the Emperor would very much like to see you, at your earliest convenience."
Gloria had been afraid of this. The Emperor's main Residence was in Rio, and Charles was bound to have been aware that she was in the city.
"My earliest convenience? Tell him that would be in about five years, give or take."
The messenger, unsmiling, shook his head. "That won't do, ma'am. His Imperial Highness was quite clear that I was not to take no for an answer. There's a limo waiting just outside, and the Emperor instructed me to tell you that there will be absolutely no media present. You can come and go in complete privacy."
"There are media reps in this room right now," Gloria pointed out. "If they see me leaving with you..."
"They will report nothing, ma'am. The Household will see to that."
And, of course, the Household could. The Empire might have had nominal freedom of the press, but in some areas, that freedom was more theoretical than actual.
Gloria bowed to the inevitable. "Let me make my apologies to the Archbishop," she said.
THE LIMO DEPOSITED GLORIA AT THE ENTRANCE to the Imperial Horticultural Gardens on the grounds of the Residence. Here, exobotanists had assembled a unique collection of plant species from all over the Empire, and had somehow gotten them to grow in harmony without annihilating either each other or the native species of Earth. The result, one critic had noted, looked as if Salvador Dali had taken up gardening.
Gloria wandered through the gateway, past a bright blue hedge, and under the drooping boughs of a quaking willow from DeSantos IV; when she brushed against its fronds, they drew back as if in fright. She spotted Charles strolling along a pathway between a fragrant, carnivorous gluetree from somewhere in the Pleiades Cluster and a pygmy sequoia that was all of four feet tall from a high-gravity world. The Emperor smiled and nodded at the sight of her, and casually walked onward in her direction.
"What's this?" he asked, pointing at her. "Clothing? On a Visitation Day? I would have expected better from an Avatar of Joy."
"It may be a Visitation Day, and this may be Rio," Gloria replied, "but back home in Manhattan, it's still January."
"Indeed. Tell me, have you had any snow there yet? The Imperial Climatologist tells me we'll see it in our lifetime."
"Not that I've noticed. But I hear they had a little up in Poughkeepsie last year."
"Something to look forward to, then. What fun, waiting for an ice age." Charles stopped in front of Gloria and held out both hands to her. She hesitated a moment, then took them in hers. They held each other and stared in silence for what felt like a long time.
Charles, at twenty-eight, was looking more like an Emperor with each passing year. His medium-length dirty blond hair was artlessly tousled over the tops of his ears, and his closely trimmed beard emphasized rather than concealed the arrogant thrust of his chin. His nose was characteristic of the Hazar Dynasty, being rather long and bony, and his watery blue eyes seemed to radiate condescension. He was still slim and fit, and was tall enough to look down on most people with lofty nonchalance.
It had been nearly a year since Gloria had seen him face-to-face. Upon her return from Mynjhino, Charles had seen fit to award her a Distinguished Service Medal; having been embarrassed and frustrated by her performance during that episode, he had concluded that the only way to deal with it was to reward her. On the other hand, her service on Sylvania had inspired no award, only icy silence.
Charles finally released his grip on her hands and gave her an appraising once-over with his eyes. She was clothed now, true, but only minimally, in a pale blue wrap dress that was loosely fastened at the waist, exposing far more than it covered.
"You've done something to your hair," he said at last.
"Oh...yes, something was done to it," Gloria admitted. Her hairdresser in Manhattan had performed some creative first aid on her damaged 'do following her return from Cartago, and her long, Dura-styled mane now flowed halfway down her back in apparent good health.
"So I heard," Charles said. "Dammit, Glory, if Mingus keeps sending you to these two-crown shit holes, sooner or later you are going to get seriously hurt. And for what? So a bunch of provincials can water their fucking lawns?"
"People live on those shit holes, Charles," Gloria responded. "They have rights and needs, the same as anyone else in the Empire. Dexta does what it can to see that they receive the respect and attention they are due."
"Spoken like the career bureaucrat you've become," Charles snorted.
"Thank you." Gloria smiled. "That's the nicest thing you've said to me in years."
Charles shook his head in evident disgust. "I cannot fathom why you are at Dexta. I never could. It's a barbaric environment, as I'm sure even you would admit, considering everything that was done to you when you started out there. And now you've become Mingus's Girl Friday, which I find appalling."
"I'm sure _you_ would."
"He's a hundred and thirty years old!"
"A hundred and thirty one," Gloria corrected. "Why, Charles, are you _jealous_?"
"Please, don't be disgusting. If you are fucking the old coot, I'd rather not know about it."
"Actually," said Gloria, "I'm not. By his choice, not mine, I hasten to point out."
"Wise of him," Charles said. "He's too old to handle an Avatar of Joy." Charles looked upward toward the treetops and the sky and shook his head. "An Avatar of Joy, Spirit save us! You know, don't you, that if it weren't for the damned Spiritists, you'd probably be spending the rest of your life on a prison world for what you did on Sylvania?"
"Me?" Gloria asked innocently. "What did _I_ do?"
"You know damned well what you did! You ruined a quadrillion crowns' worth of Fergusite. Only I can't touch you for it, because of the fucking Voice and the idiot Spiritists who think the whole thing was divine intervention, instead of a plot—a goddamn conspiracy—by you and Mingus! Don't deny it, Glory. I don't know how you managed that business with the Voice, but—"
"Oh, ye of little faith!" Gloria laughed.
Charles seemed about to respond, but stopped himself abruptly and reeled in his rage. He walked down the path a few paces and pointed at a large, gnarled tree a few meters away. "Know what that is?" he asked.
"Beats me."
"It's a glashpadoza tree. The native species on Belonna V believe that the glashpadoza absorbs all of the sins of the clan that grows it. That's why it's so hellishly ugly." He turned to look at her. "Would you care for a cutting?"
"Are you suggesting that I need my own glashpadoza tree?"
"Don't you?"
Gloria considered the question seriously. She had committed many sins on Sylvania, and the business with the Voice and the Fergusite was far from the most serious. A spaceship with five evil men in it had never reached its destination, and only Gloria knew the reason.
She looked again at the tree. "I might have a good spot to grow one of those at my place on Long Island," she said finally.
"I thought you might," Charles said. "I'll have the Imperial Gardener see to it."
Charles went over to her, stared into her face for a moment, then carefully put his arm around her shoulders and drew her closer. She didn't resist.
"I know more about what happened on Sylvania than you think I do," he said softly. "I'm not saying you were wrong. If it had been me, I'd have done the same—or worse, probably. But tell me, Glory, are you planning to make a habit of that sort of thing?"
She looked up into his eyes. "What if I am?" she asked.
"Then don't just play at it," he said. "Don't think you can go around dispensing justice and righteousness, ad hoc, on behalf of Dexta, because you can't. Sooner or later it will catch up with you, and not even Norman Mingus—not even _I_ —will be able to save you from the consequences. If you're going to do it, do it right."
"What do you mean?"
"Don't do it as a Dexta drone, Glory. Or even as an Avatar of Joy. Do it as Empress!"
GLORIA WAS TAKEN ABACK BY CHARLES'S PROPOSAL. A year and a half ago, just before she left for Mynjhino, he had made it clear that he wanted her back; but she had long since chosen another path and had no intention of retreating into Charles's waiting arms.
They had married young and divorced young. At the time, Charles was seventh in line to the throne, then sixth after old Darius finally died, and seemed unlikely ever to be Emperor. He seemed unlikely, in fact, ever to be much of anything other than a wealthy, privileged wastrel. He had persuaded her to suspend her education, and together they had spent much of their marriage gadding about the Empire in his luxurious yacht. But in her travels, Gloria had become fascinated with exosociology: the study of the lives and cultures of the Empire's many denizens, human and otherwise. When she decided to return to school and continue her studies, Charles had absolutely forbidden it. So she had left him and never looked back.
More than two years later, after Gloria had completed her studies and joined Dexta, a botched coup known as the Fifth of October Plot had killed Gregory, his two sons, and three of his nephews, leaving the Imperial throne to Charles. Gloria had sighed in relief at her narrow escape from having become Empress. The last thing she wanted was to be a useless, ineffectual ornament to the reign of Charles V. She drew immense satisfaction from her work at Dexta, and would have been miserable if condemned to spend her life in ceremonial playacting.
She broke away from Charles and began walking down one of the pathways at a brisk pace. Charles followed behind, apparently content to give her some space.
Empress! Spirit, why did he think she would want to be Empress? She had more real power now, as head of the Office of Strategic Intervention, than most Empresses ever dreamed of possessing. There had been an unbroken line of forty-seven male Emperors, dating back to Hazar the Great in 2522, 696 years ago. Unlike the kings and potentates of antiquity, producing male offspring was no hit-or-miss proposition for the Emperors of the Terran Empire. Empresses were simply the wives of Emperors, and not rulers in their own right.
Oh, five or six of them had obtained considerable power, being married to weak or dim-witted men, but they were the exceptions rather than the rule. And some Emperors had done without Empresses altogether, preferring to sire their offspring with a series of Imperial Consorts.
"I know what you're thinking," Charles said from behind her. "But this would be different. I'd give you real power, Glory."
Gloria stopped so abruptly that she skidded forward a bit on the crushed stone of the pathway. She whirled around and faced Charles.
"And what is that supposed to mean?" she demanded.
"Just what I said. Real power. We'll work out the provisions in detail, put it in writing, sign it, and publish it. I wouldn't be able to back out even if I wanted to."
She stared at him, took a couple of steps toward him, then stopped and stared at him some more, her eyes narrowing in suspicion. "What are you up to, Chuckles?"
He grinned at her. "No good, I assure you. No good at all. But since you insist, I shall reveal to you the details of my nefarious plot." Charles closed the distance between them and put his hands on her shoulders. "What I intend," he said, "is to wed and bed the most beautiful and popular woman in the Empire so that we can fuck each other like crazed minks whenever we want. And what's more, I want that woman to give me the benefit of her very great intelligence, creativity, and courage in ruling an Empire of three trillion sentient beings, every one of whom would be thrilled and delighted to have her for their Empress. There, you see? An evil plot, if ever there was one."
"You're serious, aren't you?"
"Entirely. You'd be, if you like, my coruler. I would retain primacy, of course—the laws would require that in any event. But you would have full authority in virtually any area that you prefer. Spirit, Glory, I have no interest in doing half the things I'm required to do, anyway! Believe me, I'd _welcome_ sharing the load with you. And the people would love it. Together, we'd be the best rulers this old Empire has ever seen!"
"And...?" Gloria prodded. She wasn't ready to buy into this fantasy just yet.
Charles removed his hands from her shoulders and stepped back. He nodded and said, "And...I'm under increasing pressure these days to produce an heir, one way or another. I could find an acceptable Consort easily enough, but I'd much prefer to do it with you. Host-mother, of course, so you needn't trouble yourself with a pregnancy. And it really is necessary, Gloria. I mean, you know who's next in line of succession at the moment, don't you? Cousin Larry. And I know what you think of _him._ "
"Larry?" Gloria groaned. "Lord Brockinbrough the Detestable?"
"The very one. How did you describe him that time? Old enough to be my father, unscrupulous enough to be my brother, and immature enough to be my son? Wonderful line, that. Even Larry appreciated it. Anyway, I really need to cook up a more appropriate heir, and there is no DNA I'd rather entwine with mine than thine."
Gloria frowned, looked at the ground, chewed her lower lip for a moment, then looked back up at Charles. "And...?"
"And," Charles said with a laugh, "your approval numbers in the polls are better than mine. I don't want to spend the next century competing with you, Glory. I'd rather pool our resources. And there's one more _and._ "
"Which is?"
Charles wrapped his arms around her, pulled her to him, and kissed her with Imperial urgency. She responded with urgency of her own. Time and turmoil dissolved, disappointments and frustrations faded away; whatever had gone wrong between them, some things remained as right and inevitable as they had been in the beginning. He tightened his hold on her, and she made no attempt to resist. Moments later they lay sprawled in the grass beneath the gnarled limbs of the glashpadoza tree.
IN MIDAFTERNOON, GLORIA TRANSITED BACK to Dexta Headquarters in Manhattan, after spending two hours rolling in the grass beneath the glashpadoza tree with the Emperor. Charles had genetic enhancements of his own, and despite his essentially selfish nature, he was a marvelous lover. Whatever regrets Gloria had about their marriage, they didn't include sex.
She walked along the main concourse, feeling every eye on her, as if people somehow knew she had just been humping the Emperor. But what an afternoon...
Empress! Spirit!
She had made no commitment, but she hadn't said no, either. That, in itself, surprised her. It was an offer that could not be dismissed casually, if at all. Certainly, it would be a life-changing decision. Charles understood that and didn't press the issue; they had agreed that both of them would give the matter more thought.
Gloria got into an elevator with several people she didn't know and said, "Forty," where the OSI had its offices. She heard two women standing behind her giggling to each other and resisted an urge to turn and look.
Another woman said, "Congratulations, Ms. VanDeen." Gloria's eyebrows shot upward. How could she have known?
Sensing Gloria's confusion, the woman added, "About being selected as an Avatar of the Spirit, I mean. That's quite an honor."
"Oh, yes, of course. Thank you."
"It reflects well on Dexta," the woman said. "We're all very proud of you." There were more giggles from behind her. The car reached the fortieth floor and Gloria smiled good-bye and stepped out, feeling relieved. Between Charles and the Church, Gloria was feeling strangely vulnerable and conspicuous today.
She entered the outer office of the OSI and stood just inside the door for a few seconds, like a queen bee surveying her hive. After a slow and uncertain start, the Office of Strategic Intervention was now a going concern, vibrant and humming with activity and purpose. They had handled five interventions, which had required Gloria's personal presence on the affected worlds, and were now ratcheting up their operations to include a host of lesser matters that could benefit from the OSI's attentions. Gloria had soon realized that she couldn't go everywhere and do everything herself, and there was really no need. She had assembled a crackerjack staff, and her people relished their assignments.
She smiled to her troops as she made her way to her own office and accepted their congratulations. Guarding the door to her inner sanctum, as usual, was her Executive Assistant and best friend, Petra Nash.
"Good afternoon, Your Avatarness," Petra said. "How does it feel to be a holy-holy?"
"Just fine," Gloria answered. "Anything happening that I need to know about?"
"A couple of messages on your console, nothing real urgent. Grant Enright is still in Bombay with the GalaxCo people, and Althea is off to Luna for a long weekend with someone whose name I'm not supposed to mention, so I can't tell you that it's the Duke of Glastonbury. Jill Clymer says she'll have a report on some potential mess in Sector 19 for you tomorrow, and, let's see, Phil Benz wanted me to remind you that he's got Naval Reserve duty coming up next week. And if you have a few minutes, Pug and I need to talk with you."
"Certainly. Just come on in."
"Will do. Give me a minute to find the Pugnacious One."
Gloria smiled and watched for a moment as Petra got up from her desk and went off in search of Pug Ellison, her assistant, roommate, and lover. Petra was positively glowing these days, and Gloria could only marvel at the way her friend had blossomed with newfound self-confidence over the past six months. At Gloria's urging and with Pug's ardent approval, Petra was trying to be less of a Dog and more of a Tiger, and seemed to be managing the transition with style and enthusiasm. Today, she was showing her Tiger stripes in a gray miniskirt that barely concealed her shapely bottom and a matching jacket buttoned only at the waist to show off the subtle curves of her small, pert breasts.
In the metaphorical (but quite real) menagerie of Dexta, there were Lions, Tigers, Dogs, Moles, and Sheep. The Lions, mostly male, were the natural leaders, relying on strength and force of personality to secure their positions. Tigers, mostly female, were sleek and beautiful, using sex the way Lions used strength. Dogs came in two breeds: Pack Dogs, who roamed the lower levels of the bureaucracy as feral predators, savaging the weak and unprotected, and Lap Dogs, whose loyalty won them the patronage and protection of a willing superior. Moles were sneaky bureaucratic infighters, and the numerous Sheep were the anonymous, workaday backbone of the system.
The species had evolved over the centuries because Dexta was, by design, a Darwinian jungle where only the strong and smart could hope to survive and flourish. Each year, a hundred thousand new Level XV staffers joined Dexta, and fully 20 percent of them failed to survive the first twelve months. The Fifteens—and, indeed, everyone else, to one degree or another—were subjected to every imaginable form of social, psychological, physical, and sexual abuse in the brutal environment of Dexta. The point of it all was to weed out the weaklings and assure that those who remained functioned at peak efficiency all the time. The system was cruel and often downright sadistic, but it worked, and that was all that really mattered.
Both Gloria and Petra had almost failed to survive that first dreadful year. Gloria managed by becoming a Tiger—predator rather than prey—and Petra had become her faithful assistant and Lap Dog. Gloria was, with her background, breeding, and beauty, a natural Tiger in any case, while Petra—a diminutive, clever, but insecure refugee from an impoverished home in nearby Weehawken—had taken a while to work out for herself just who and what she was and wanted to be. But she seemed to have hit her stride finally, and Gloria was happy for her.
Gloria sat down at her desk, checked her console for a few moments, then looked up as Petra arrived with Pug in tow. Pug—Palmer, formally—was a good-looking young man with brown hair, blue-gray eyes, and open, friendly features. He was a bit under medium height, which left him a good seven or eight inches taller than Petra. And, at twenty-four, nearly three years younger. He was a Level XIV to Petra's Level XIII, but didn't seem to mind being her assistant. He was, in fact, extravagantly grateful to Gloria for bringing him aboard as a permanent member of the OSI team.
She had originally recruited him for the mission to Sylvania the previous spring, when OSI needed a band of independently wealthy, big-bucks bureaucrats who could stand up to the get-rich-itch that seemed to infect everyone who came to that brawling boomworld. Pug, a Level XV at the time, had been eager to prove that he was more than just a member of a fabulously wealthy family from New Cambridge, and his work had impressed Gloria. He and Petra had caught each other's eyes, and eventually they wound up in each other's arms.
Back in Manhattan, Gloria had used her clout as the Level X head of OSI—and her personal wealth—to secure a sumptuous apartment for Petra and Pug in the same midtown building as her own glamorous penthouse, three blocks from Dexta HQ. Normally, Thirteens and Fourteens resided in designated Dexta quarters in Brooklyn, but Gloria felt she owed Petra something better after their perilous and painful mission on Sylvania. For Petra, who'd grown up in dire poverty, the apartment seemed like a palace; for Gloria, who'd grown up in an actual palace, the apartment seemed a pittance. Gloria still felt guilty about what had happened on Sylvania, even though Petra, blessedly, had no coherent memories of the rape and assault that had nearly killed her.
Gloria got up from her desk and went over to the couch, beckoning Pug and Petra to join her there. Pug first got them coffee and tea, while Petra leaned close and inspected Gloria's new ruby mustard seed. "That's gotta be the smallest gem I've ever seen you wear," she said. "Usually they're about the size of softballs."
"We Avatars are a modest lot," Gloria explained. "So, what did you want to see me about?"
"Cartago," said Pug as he handed her a cup of coffee, then sat next to Petra on the couch. "We've gotten the initial report back from IntSec, and it's a little troubling. The Bugs have also turned up some additional information that—well, I'll let you characterize it for yourself. But it's a bit strange."
"Make that a _lot_ strange," Petra amplified.
"Oh? Let's hear it."
"Well, first of all, the shooter on Cartago," said Pug. "There wasn't a lot left of him, and the DNA trace was inconclusive. That's not too surprising in itself. I mean, there are three trillion people in the Empire, and we don't have definitive genotypes on all of them. But the analysts do say, with a ninety-five percent confidence level, that the shooter was not a native of Cartago. And in light of what we've learned about the weapon...Well, as I said, it's troubling."
"What about the weapon?"
"It was a Mark IV plasma rifle."
"A Mark _IV_?" Gloria asked. "I thought those were obsolete."
"Not obsolete, exactly," Pug said. "I mean, they still work. But they've been outdated for about thirty or forty years. But this particular weapon is of special interest. You see, according to the serial number, it was a part of a shipment to Savoy at the beginning of the war with the Ch'gnth."
"Savoy? You're kidding!"
Pug shook his head. "The numbers check out. According to the records, that particular weapon was manufactured on Ostwelt in June of 3163, and shipped to Savoy that September, just a week before the Ch'gnth attack."
"Savoy!" Gloria exclaimed, impressed by the very mention of the name. To a denizen of the early-thirty-third-century Empire, saying that the rifle had come from Savoy was like saying it had come from Waterloo or Gettysburg. The Empire's last great war, against the Ch'gnth Confederacy, from 3163 to 3174, had been a desperate struggle, and it had all turned on the outcome of the first crucial battle for Savoy, in September 3163. The Empire's garrison on Savoy—and the entire population of the colony—had been wiped out by the Ch'gnth, but their last-ditch resistance had bought crucial time that allowed the Imperial Navy to assemble a fleet. It had struck back at Savoy in one of the most decisive naval engagements in history. Military historians compared it with Salamis and Midway. It had turned the tide of the war and led to the eventual Imperial victory.
"So," Gloria said, "we have a shooter who's not from Cartago using a weapon that should have been destroyed fifty-odd years ago. And for some reason, he wanted me dead."
"Initially," Pug said, "the Bugs figured that he may have been connected to PAIN or PHAP, but he could just as easily have been working a zamitat contract, or just some freelance nutcase." PAIN—the People's Anti-Imperialist Nexus—was an anarchist terrorist organization of marginal efficiency, and PHAP—the Pan-Human Alliance for Purity—was a racist fringe group of doubtful sanity. The zamitat was an Empire-wide criminal network, with ancestral ties leading back to the Mafia, the Yakuza, and similar organizations on other worlds. Gloria couldn't imagine why any of them would have targeted her.
"I can't see PHAP," Gloria said. "No aliens on Cartago. PAIN, you can never tell about, but they don't usually go in for solo assassinations. And I don't think the zamies have anything against me."
"Maybe not," Pug said, "but there's something else."
"This is the troubling part," Petra said.
"Which is...?"
"This," said Pug, "is the second Mark IV plasma rifle from the Savoy shipment that has turned up in the past week."
"Last night," Petra said, "we received a report from Watami III, in Sector 23. Six days ago, there was a terrorist attack on the Dexta offices there. They killed seven of our people, including the Imperial Secretary, along with three civilians who happened to be in the office. The complete list of victims is in your console."
"Spirit!" Gloria said, genuinely shocked by the news. "What happened?"
"Details are still a bit sketchy," Pug told her, "but it seems that three people burst into the office and opened up with plasma rifles. An Internal Security man killed one of them, but the other two made a clean getaway. Apparently they just fired at random, killing anyone they could. An hour later, PAIN released a statement on Watami, claiming credit for the attack."
"And then," Petra said, "this morning we got a report of another attack, four days ago on Kyushu Prime in Sector 20. Same sort of thing—three attackers with plasma rifles shooting up the Dexta office. Only four dead in this one—two Dexta, two civilian. And all of the attackers got away. Again, PAIN claimed it as their work."
Gloria leaned back on the couch, a deep frown creasing her features. "What the hell is going on?"
"IntSec is still putting it all together," said Pug. "Volkonski says he'll have a report for you in a couple of hours. But here's the thing that Petra and I are worried about. The terrorist who was killed on Watami was also using a Mark IV plasma rifle. And that rifle was also part of that Savoy shipment of 3163."
Gloria silently stared at Petra and Pug for several seconds.
"When Arkady gave us the initial report from Internal Security," Petra said, "he was very insistent about scheduling a meeting to set up coverage for you." Arkady Volkonski, head of the OSI's IntSec section, took it as his life's mission to see to it that nothing bad ever happened to Gloria or anyone else in OSI. He was, in Gloria's view, a trifle overzealous about it.
"He thinks," Pug said, "and we agree, that PAIN was also behind the attack on you on Cartago. From their point of view, it would make sense. They hate Dexta and the Empire, and you are a prominent symbol of both."
"Yes, but—" Gloria started to protest, then broke off. She didn't want to accept the notion that PAIN had specifically targeted her, but the evidence was staring her in the face. "The weapon," she said with glum resignation.
Pug nodded. "The Bugs don't have a clue how those weapons suddenly turned up, but they clearly connect Cartago with the other two attacks. Volkonski says you need protection, and Petra and I agree."
"I'll see him tomorrow," Gloria said, hoping she could find a reason between now and then to avoid it. She didn't like the fact that the crazies in PAIN wanted her dead, but she was also not about to submit to the round-the-clock attentions of the Bugs.
"Gloria?" Pug leaned around Petra to look at her.
"What?"
"Your safety is our primary concern, of course," he said. "But there's also a larger issue here."
"The weapons," Gloria said.
"Exactly. How in hell are weapons that were supposedly lost more than fifty years ago on Savoy popping up now? How did PAIN get their hands on them? And what happened to those weapons in the first place, back in 3163? We're going to need to do some serious historical research, and not all of the records we need are on Earth."
"Where are they?"
"Well," Pug said, "you know that things used to be organized a little differently. I mean, back before the war, before Secretary Mingus took over and reorganized the Department. Quadrant Administration used to be out in the field, not here in Manhattan. Savoy is in Quadrant 4—and both attacks on Dexta offices were also in Quadrant 4—so we'd need to go to the actual site of the old Quad Administration offices to get at the original records."
Gloria frowned. "I asked a simple question," she said. "For some reason, I'm not getting a simple answer. Just where is the actual site of the old Quad Administration offices?"
"Well..."
"On New Cambridge, Gloria," Petra broke in, rescuing Pug from further circumlocutions.
"Aha," Gloria said.
Pug smiled in embarrassment. "I was afraid you might think..."
"That you wanted an excuse for a little working vacation on your homeworld?"
"Um, well..."
"We'll have none of that in _my_ office," Gloria declared firmly. "If it is necessary to send someone to New Cambridge to investigate this matter, they will go there to _work._ Is that understood?"
"Yes, ma'am."
"Good. Now, I think this does deserve further investigation, and I think it makes sense to send someone who is already familiar with New Cambridge and has connections there. I suppose that means you, Pug, but I'm reluctant to send a Fourteen off on an important field assignment without proper supervision. I think you need at least a Thirteen along to make sure you keep your nose to the grindstone and don't fritter away your time on family reunions and sightseeing tours and romantic getaways. Petra, I suppose I can rely on you to see to all of that?"
"Absolutely!"
"I knew I could count on you. Now, it occurs to me that this year's Quadrant Meeting is in Quad 4, and it's scheduled for New Cambridge in a few weeks. I wanted to get to that, myself. Ever been to a Quadrant Meeting?"
Both Pug and Petra shook their heads. "I went to one three years ago," Gloria said, smiling at the memory. "Quite a show."
Each year, on a rotating basis, one of the Quadrants hosted a grand gathering of the tribes. In an empire where communications took days or weeks, it was important for Dexta people to have a chance to make personal contact with their far-flung coworkers. Conflicts and controversies could be resolved face-to-face at the Quadrant Meetings, and they gave the Dexta brass an opportunity to assess the mood and morale of their troops. The formal agenda at the meetings always included a stupefying round of panels, committees, speeches, and seminars. Informally, much else happened.
"It's like going to the circus," Gloria said. "Anyway, I think the two of you should leave as soon as possible in order to have some results to report by the time I get there. I'll get a Flyer for you. Can you leave, say, the day after tomorrow?"
Pug and Petra grinned at her. "Anything you say, Gloria," Pug said.
"Uh...there is one thing, though," Petra added. "If we're going to be digging around in official Dexta records for the Quadrant—I mean, when we aren't frittering away our time with family reunions, sightseeing tours, and romantic getaways—we really ought to get authorization from the Quadrant Administrator."
"Good idea," Gloria said. "And while we're at it, another thought occurs to me. You said Jill Clymer has something brewing in Sector 19? That's Quad 4, too. Get her in here, then get me an appointment for this afternoon, if possible, with Cornell DuBray."
"Wow," said Petra. "The Quad Admin himself? Level Four?"
"We Avatars of the Spirit," Gloria said grandly, "don't bother with mere underlings. Anyway, I've been looking for an excuse to get acquainted with the upper-level Eagles, and this will do." She got to her feet and walked to her desk.
"Uh, Gloria...?"
"What, Petra?"
"I think you ought to change before you go see DuBray."
"Why? What's the matter with what I've got on?"
Petra walked over to Gloria and reached for her back. She plucked something off it and held it up for Gloria to see. It was a blade of grass.
"And there are some grass stains, too."
Gloria rolled her eyes. No wonder people had been staring and giggling behind her back all afternoon! And that bastard Charles hadn't said a word...
"Life's just one big roll in the grass for you Avatars, huh?" Petra grinned impishly at her boss.
"Do not deny yourself joy," Gloria intoned dryly.
"Don't worry, we won't. And thanks, Gloria!"
GLORIA ENTERED THE OFFICE OF QUADRANT 4 Administrator Cornell DuBray dressed, she felt, like a proper Avatar of Joy. She had changed to a white lace band skirt—just a hoop of all-but-transparent fabric four inches wide—slung low on her hips. Her breasts were similarly covered, but not at all concealed, by a narrower hoop of the same fabric. She didn't normally come so close to outright nudity in the office, but it was, after all, a Visitation Day. And, in any case, she wanted to make a strong first impression on DuBray.
Far above the lowly Lions, Tigers, and Dogs in the Dexta pecking order were the Eagles—Level VII's and above—who actually ran the organization. Normally, a Level X like Gloria would never have any contact with a Level IV like DuBray. But the Charter of the Office of Strategic Intervention, written by Norman Mingus himself, gave Gloria broadly defined powers and responsibilities and virtually unlimited access to Dexta personnel and records.
She had checked DuBray's file before coming to his office to gain a little more insight into a man who was already something of a Dexta legend. Cornell DuBray—ninety-seven years old and still as handsome and vigorous as a man half his chronological age—had been Quad 4 Admin for nearly forty-two years. For purposes of comparison, Gloria noted that DuBray's fellow Quadrant Administrators had been at their respective posts for twenty-three, nine, and six years. He had taken over the slot from Norman Mingus in 3176, when Mingus became the Dexta Secretary. He was the odds-on favorite to succeed Mingus as Secretary when Mingus finally retired or died—assuming he ever did.
During his tenure in Quad 4, DuBray had built a reputation as a tireless, ruthless, and sometimes tyrannical Administrator who never forgot a friend or a foe, and whose rewards for the former and punishments for the latter were equally lavish. There was nothing about it in the files, but it was said around Dexta that some of DuBray's early opponents had wound up on high-gravity prison worlds; there were, apparently, no recent opponents. DuBray's loyalty to Norman Mingus was unshakeable, and it was reciprocated by the Dexta Secretary.
He greeted Gloria at the door and ushered her into his palatial 110th-floor office. DuBray was just over six feet tall, beefy but not fat, and had long, pompadoured silvery locks and a thin mustache to match. His nose suggested Hazar blood, and his full lips looked supple and feminine. He had been married four times—leaving him one wife shy of Norman Mingus—and was said to pursue a sex life that the Sept of Joy might have noted with approval. He clasped Gloria's hands in his and stared at her, head to toe, with undisguised appreciation.
"Ms. VanDeen," he said, "it is a true pleasure to meet you at last. I suspected and hoped that our paths would cross eventually. Norman speaks of you often, and holds you in the highest regard. And I believe congratulations are in order today, are they not? I don't know that Dexta has ever had its own Avatar of the Spirit."
"Actually, there have been three," said Gloria, who had checked, "although it's been two hundred years since the last one. But thank you, Mr. DuBray. I've been looking forward to meeting you, as well. I had hoped to make the rounds of all the Quadrant Administrators long before now, but OSI has kept me pretty busy."
"So I gather," said DuBray. "Can I get you something to drink? Some wine, perhaps?"
"That would be lovely," Gloria said. She took a seat on a divan facing the windows, giving her a magnificent view of the January sunset over New Jersey. DuBray joined her on the divan a moment later, gave her a goblet of red wine, and clinked glasses with her.
"To Dexta," he said. "And to our mutual friend and patron, Norman Mingus."
"Long may he wave," Gloria said. She took a sip of the wine and found it excellent.
"A cabernet from the vineyards of Sonoma III, vintage 3196. The finest wine in my Quadrant and, for my money, the entire Empire."
"It's marvelous," Gloria agreed, taking another sip.
"I'm glad you think so. I'll have a case of it sent to you."
"Oh, please, don't trouble yourself."
"It's no trouble at all, I assure you," DuBray said, smiling wolfishly. "I didn't say I'd deliver it myself. Although I would be happy to do just that. You truly are a remarkably beautiful woman, Ms. VanDeen. I've seen the vids, of course, but having you here—in the flesh, as it were—adds an entirely new dimension to my appreciation of your charms."
"That's very nice of you to say, Mr. DuBray. And please, call me Gloria."
"Very well, I shall. Now tell me, Gloria, what good fortune brings you to me today? What can I do for you?"
"Well, as I said, I had hoped to make your acquaintance in any event, but I do have the excuse of some business matters. I wanted to let you know the OSI is going to be doing some work in Quadrant 4."
"That's very thoughtful of you. And what desperate shortcoming in my Quadrant, might I ask, requires your strategic intervention?" DuBray's eyes had narrowed slightly, and the look on his face was not entirely pleasant. This was a reaction Gloria had seen before from planetary and Sector authorities. If an OSI intervention was necessary, it could only mean that there was a malfunction somewhere in the existing Dexta bureaucracy. No one was happy to hear that.
"Oh, nothing desperate, I assure you," Gloria said as lightly as she could. "But it does look as if there have been some financial irregularities in Sector 19. One of my people is preparing a full report, and I'll send you a copy tomorrow. It only came to my attention this afternoon, so I'm afraid I don't have many details yet. But, in general, it seems that there has been some double- and perhaps even triple-flagging of various freighters. If the records are accurate, some freighters have apparently been showing up in two or more ports at the same time. We aren't sure exactly what's going on, but of course there are tax implications and the possibility of hijackings and phantom loads."
DuBray nodded. "I see," he said, and took a swallow of the excellent wine. "And what does Sector Administration have to say about all of this?"
"Nothing, yet," said Gloria. "Since we aren't sure just who or what is involved here, we thought we'd hold off on notifying Sector. Under the circumstances, we felt it might be best to go directly to Quadrant."
"And swoop down on the malefactors from above, without warning?"
"If necessary. Of course, at the moment we still don't even know if there _are_ any malefactors. It could all turn out to be accounting errors, or something similar. But to be safe, it made sense to handle it through Quadrant, initially."
"Indeed." DuBray pursed his lips, crossed his legs, and stared beyond the slums of New Jersey toward the setting sun. "And you will be going to Sector 19 yourself?"
Gloria shook her head. "An OSI team has already been dispatched, but I have no immediate plans to get personally involved. There is, however, another matter..."
"More happy news regarding my Quadrant? My cup runneth over."
Gloria shrugged. "Sorry, but part of my job is to tell people things they'd rather not hear."
DuBray looked at her again. "And you dress this way to soften the blow?"
"If it has that effect," Gloria said, "then so much the better. But the truth is, I dress this way because I enjoy it, and I've noticed that other people generally enjoy it, as well. You certainly have."
DuBray offered her a smile. "Point taken," he said. "Very well, then. Proceed."
Gloria took a deep breath, then launched into it. "Mr. DuBray," she said, "I'm sure you've already heard about the two PAIN attacks in your Quadrant."
DuBray nodded. "Ugly," he said. "Very ugly. But Internal Security is already dealing with it. What is OSI's interest in this?"
"One of the terrorists employed an old Mark IV plasma rifle," Gloria said. "And last week on Cartago, another Mark IV was used in an attempt on my life. Both of those rifles were part of a shipment to Savoy in September 3163, just before the start of the war with the Ch'gnth."
DuBray gave her a sharp, probing stare, then abruptly got to his feet. He gazed at the glow on the horizon for a moment, drained the rest of his glass in one long pull, then went over to a sideboard to take the bottle and refill his glass. "Interesting," he said finally, after taking another sip of the wine.
"We need to get at the original records from 3163," Gloria continued, "when the Quadrant Administrator's office was on New Cambridge. Two of our people will be leaving for there in a couple of days. Of course, the OSI Charter gives them all necessary powers, but things might go a little easier for them on New Cambridge if they could take along an authorization from you."
DuBray took another swallow of wine and turned to look at her. "There were those of us," he said, "who warned Norman that this Office of Strategic Intervention was a spectacularly bad idea. Dexta already has an Inspector General's Office and a Comptroller, after all, and they have managed to keep the gears and cams reasonably free of grit down through the centuries, without any 'strategic interventions.' Some of us, in fact, told him to his face that his obvious infatuation with a certain young woman was leading him to make a potentially disastrous decision. But, of course, he ignored our advice—as was his privilege."
Gloria got to her feet and faced DuBray. "I see," she said.
"I doubt it. Come over here, if you would, Gloria." After locking eyes with him for a few seconds, Gloria walked around the divan and approached DuBray. He regarded her frostily for a moment, then put his glass down and, with no preamble, unknotted her top and pulled it away from her. Then he looked down at her bare breasts and smiled crookedly.
"What do you think you're doing?" she demanded.
"Isn't that obvious? I suppose you could say I'm performing a little strategic intervention of my own. If you want to play your games in _my_ backyard, little girl, you'll play by my rules, or you won't play at all."
"We'll see about _that,_ " Gloria replied with some heat. "I'm not some bloody Fifteen anymore, and I don't have to put up with this bullshit from you or anyone else now! I'm head of the OSI and a Ten!"
"Yes, and I'm a Four," DuBray said blithely. "You see where that leaves you, don't you? Honestly, Gloria, did you really believe that things were any different at the upper levels of Dexta than at the lower ones? We just play the game with a little more finesse and style up here. When you were a Fifteen, you saved your job by letting the Pack Dogs fuck you on the floor of the restrooms, didn't you?"
Gloria could only stare at him in smoldering silence. She wondered how in hell he had heard about _that._
"Well," DuBray went on, "I have a very comfortable bed in the next room. If you expect to strategically intervene in my Quadrant, you'll join me on it."
The idea of having sex with a superior at Dexta to preserve or promote her career was hardly novel to Gloria. That had always been part of the game, and by becoming a Tiger she had committed herself to playing it. But after Mynjhino and her appointment to head the OSI, she had come to realize that her success at Dexta depended on the quality of her work more than on her sexual stratagems. And there was no one at Dexta—not even Norman Mingus—that she _had_ to screw. So she was annoyed by DuBray's arrogance and presumption.
Gloria smiled sweetly at him. "I'm afraid I'll have to decline your very kind invitation, Mr. DuBray. You might have had a shot at me if you actually had half the finesse you think you have. But you see, I'm allergic to assholes."
Gloria pivoted smartly and marched out of the office, leaving DuBray standing there holding the scrap of cloth in his hand.
"You're making a mistake," he said over her shoulder, but Gloria just slammed the door on him.
"THAT MOTHERFUCKING BASTARD! THAT INCREDIBLE son of a bitch!"
Gloria swept into Petra's apartment like a thunderstorm, raging almost incoherently as she paced around in little circles, her thin lips drawn tight and her eyes blazing. Petra didn't think she'd ever seen her so overwrought. Wearing nothing but a tinier-than-usual band skirt, her ruby mustard seed bouncing around between her breasts in her agitation, Gloria seemed to be on the verge of doing physical violence to someone. She usually reserved language like this for Charles—but somehow, Petra didn't think it was directed at him this time.
"Which motherfucking bastard?" Petra ventured to inquire.
"DuBray! Cornell Fucking DuBray, that arrogant asshole! Who the hell does he think he is, treating me like a goddamn Fifteen?"
"What happened?" Pug asked.
"Nothing happened! Nothing at all, because I didn't _let_ it happen! I'm a _Ten_ now, dammit, not some frightened, brainless Fifteen! How _dare_ he treat me that way? How _dare_ he? 'If you want to play games in _my_ backyard, little girl, you'll play by _my_ rules!' Can you imagine? He actually said that! _Little girl?_ Why, that arrogant, motherfucking—"
Gloria stopped short when she finally realized that she, Petra, and Pug were not the only people in the room. Petra's mother stood at the doorway to the kitchen, gazing at Gloria with more than her usual degree of disapproval.
"Uh...Mrs. Nash!" Gloria stammered. "I didn't see you. Uh...hello. How are you? I'm sorry, I didn't...uh, I apologize..."
"Mom," Petra quickly broke in, "Gloria's had a tough day at the office."
"Yes, I can see that," said Mrs. Nash. "Perhaps she should have stayed there until she recovered some semblance of civility."
Petra sighed. Her mother had never approved of Gloria, or her daughter's association with such a person. Oh, she didn't mind the perks that went with knowing Gloria, like the invitations to fancy parties in Gloria's penthouse, or the free family vacation Gloria had sprung for last year; but she had always found Gloria herself to be too wild and disrespectful of propriety. In particular, she disapproved of Gloria's repeated public contretemps with the Emperor. In Mrs. Nash's view, the Emperor—any Emperor—ought to have been treated with the utmost deference and respect. Of course, Mrs. Nash had never even met an Emperor, let alone been married to one.
The only man Mrs. Nash had ever been married to in her sixty-three years was Mr. Nash, Petra's father, who had fled some twenty years ago, when Petra was six. In consequence, she distrusted men in general—much as she distrusted women who consorted with them. She tolerated Pug, principally because he was from a wealthy family, but somehow always managed to convey the impression that she considered him to be beneath her daughter. Her mother, Petra realized, was a galaxy-class snob, probably because she had been poor all her life. Now that her daughter was moving up in the world, it gave her new opportunities to express her disapproval of and disappointment in virtually everything and everyone, including Petra.
"Gloria," Petra said, "Mom's going to be minding the apartment while Pug and I are on New Cambridge. Uh...we're still going, aren't we?"
"Damn right you are...I mean, yes, of course. You won't have DuBray's seal of approval, which may mean that you'll have to do without any cooperation from the Dexta staff on New Cambridge. But you've got full authority under the OSI Charter, so don't put up with any bullsh—uh, don't—"
"We won't," Petra said. "Can I get you a drink or something?"
"No, no, I just wanted to let off some steam on my way upstairs." Gloria turned to face Mrs. Nash and said, "I do apologize for my unseemly outburst, Mrs. Nash."
"That's quite all right, Gloria," Mrs. Nash said loftily, now that she had attained the moral high ground. "Tell me, though, do you usually dress like this at work? From the way my daughter dresses these days, it wouldn't surprise me, but I confess, I didn't think you went around the office _completely_ naked."
"Well," Gloria tried to explain, "it's been kind of an unusual day, Mrs. Nash. You see..."
"It's a Visitation Day, Mom," Petra put in. But that didn't impress Mrs. Nash, who had sampled and discarded Spiritism years ago.
"Yes," she said, "I believe I heard something about that vulgar... _ceremony._ I gather you are now some sort of Avatar? I suppose that means you can go around naked all the time, if you want?"
"Well, as a matter of fact, it does," Gloria said. "But to tell you the truth—"
"That's quite all right, Gloria," Mrs. Nash cut in. "You owe me no explanations. I don't pretend to understand you young people in any case. However, I can tell you that when I was your age, we showed a little more respect for—"
"Bullshit, Mom!" Petra had heard about as much as she could take. "When you were our age, old Darius was on the throne and people acted a lot wilder than anyone does today. So spare us your laments for the Good Old Days, okay? Gloria, I'm sorry."
"No, I'm the one who's sorry. I just needed to vent. See you at the office tomorrow. Good night, Pug...and Mrs. Nash." Gloria made a hasty exit.
When Gloria was gone, Petra, fists on hips, confronted her mother. "After all Gloria's done for us," she said, "you'd think you could make at least a little effort to be nice to her!"
"After what _she's_ done?" Mrs. Nash said in frank astonishment. "You mean, after all _you've_ done for her! Why, that woman could hardly function without you. And after what happened to you on Sylvania—"
"I've told you before, that wasn't Gloria's fault."
"No, of course not. All she did was force you to work as a prostitute. And if a prostitute gets assaulted, why, it hardly matters, does it?"
"Dammit, Mom, you know perfectly well I was working undercover as a bar girl, _not_ a prostitute!"
"Well, I suppose the distinction escapes me. I tell you, that woman has corrupted you, Petra. Yes, that's it, she's _corrupted_ you! Just look at the way you're dressed right now."
"Yeah," Pug said, grinning.
"You keep out of this, young man. I know where _you_ stand in all of this. People like you and Gloria just seem to think you can _buy_ whatever—and whomever—you want."
"Right," said Petra. "Like Gloria bought us this apartment."
"Gloria? I was under the impression that your young man paid the rent."
"Pug pays the rent, but Gloria got us the lease. It would have taken years to get a place half this good without her help."
"Well, that was very nice of her, then, I suppose," Mrs. Nash allowed. "Of course, the way she works you, you barely get a chance to spend any time here, as it is. Fortunately, you have a mother who is willing to drop everything at a moment's notice and take care of your precious apartment for you."
"Drop what?" Petra wondered. "Tuesday night bingo in Weehawken? The two-crown window at Paramus Raceway?"
Mrs. Nash tossed her nose into the air and sniffed, "I don't have to stand here and be insulted." With that, she swiftly retreated into her guest bedroom.
"Good," Petra said loudly enough for her to hear, "then I won't have to stand here and insult you!" Petra shook her head and turned to face Pug. "Mothers," she cried.
Pug grinned at her. "Wait till you meet mine."
GLORIA FOUND A CASE OF SONOMA III CABERNET, vintage 3196, waiting at her front door when she reached her penthouse. She briefly debated whether to dump it all down the garbage chute but decided that would be a waste of outrage—and good wine. Instead, she took the case inside, opened a bottle, drank a glassful as she fixed herself a salad, and then another as she ate it. DuBray might have been an asshole, but he knew his wine.
She supposed she should have expected it. DuBray certainly had the reputation. Still, it had been more than two years since the last time a superior at Dexta had insisted that she have sex with him. Somehow, she had simply assumed that she was beyond all of that now. As a notorious Tiger, she had established a reputation of her own, and even before Mynjhino and her promotion, the Twelves and Tens she had dealt with on a daily basis had shown her a wary respect. If Gloria had sex with anyone at Dexta, it was by her choice and on her terms. Until now...
Gloria felt like smashing something. But she returned her salad plate to the kitchen intact and poured another glass of wine instead. Rather than smashing something, she decided, she would _get_ smashed. After a day like today, she figured, she had earned it.
She was so pissed off about DuBray, she had hardly even thought about Charles and his offer. Empress—with real power! That suddenly had great appeal; as Empress, she might have DuBray flayed alive, inch by inch. There were, she knew, catacombs deep beneath the Residence in Rio where such things had actually happened—centuries ago, supposedly. She doubted that Charles had ordered any tortures himself, although he was such a bastard at times that she wouldn't rule out the possibility.
Gloria took the bottle and glass and curled up on a comfortable sofa. "Music," she said. "File C, random." A moment later, the penthouse reverberated with the raw rhythms of Early Blues. Bessie Smith, Muddy Waters, Etta James, B. B. King—Gloria knew and loved them all, and when she'd had a little more wine, she began to sing along.
A little more wine found her dancing as well as singing. Just like a good little Avatar of Joy.
Her mother would be thrilled when she heard the news. Gloria's parents were fabulously wealthy, truly wonderful people, and if they couldn't quite understand why their daughter should want to hold down an actual job—much less one at Dexta—they at least tolerated her attempts to make herself a useful and productive member of society. They had apparently never felt the need to be anything but what they were—a yachtsman of Empire-wide renown and his devoted mate. Currently, they were six hundred light-years away on some watery world for a round-the-planet regatta. Dad knew everything there was to be known about the sea, and used to tell her stories about Nelson and Magellan and Cook. And Mom was loving, smart, and sexy; she would be so proud when she heard that Gloria was an Avatar of the Spirit. She'd be even prouder if Gloria accepted Charles's offer and became Empress. In her own way, Gloria's mother was just as much of a snob as Petra's.
Fred and Georgia VanDeen were such good people; Gloria missed them desperately at times. Petra's mother might have been a pain, but at least she was handy. Sometimes Gloria wanted to cry when she thought about how far away Mom was, and how long it had been since she had been able to crawl into that warm, loving lap and tell her all her troubles...like...like how people were trying to _kill_ her, and how they'd missed by _that_ much! How she could still smell the burned hair...
Ray Charles was crooning about how Georgia was on his mind...and Mom—Georgia—was on Gloria's mind, too, and she wanted to be able to tell her about...about... _the burned hair_!
_Spirit! If she'd bent to take her seat a tenth of a second later..._
And suddenly, Gloria was sobbing and shaking. She couldn't stop. She tried to pour some more wine, but just couldn't manage it. Couldn't stop shaking.
_Mom, they're trying to kill me!_
First there was Mynjhino, when she'd had more narrow escapes than she could count. And then Sylvania, where no one had actually tried to kill her, but Grunfeld's thugs had tried to rape her...and then, all the burned and ruined bodies at Pizen Flats...Ted, Gus, all of them. Burned the way she would have been burned on Cartago if she'd been just a fraction of a second later...
Gloria sank to the carpet and curled herself into a ball, but still she could not stop shaking. It all overwhelmed her now—all the close calls, the people who depended on her to do her job, the faith of Norman Mingus, the trust and love of Petra, the confidence of her staff, and Charles, yes, Chuckles himself, back in her life suddenly, and he still wanted her, and that bastard DuBray wanted her, too, an Avatar of the Spirit now, and all of it, all of it, pressing down upon her, and Georgia on her mind and so far away... _and the smell of burned hair!_
THE LIMO SKIMMER MADE ITS WAY UPTOWN, toward Harlem. For centuries, Harlem had been the home of the city's African community, and to the extent that there was still such a thing as an African community in Manhattan, it was still in Harlem. But after more than a millennium of Transit technology and global culture, distinct ethnic communities were getting hard to find on Earth. For anything resembling ethnic purity, it was necessary to travel to the colonies, where physical isolation and (in some cases) intentional segregation had preserved a degree of cultural and genetic distinctiveness. In Manhattan, it was easier to find Chinese restaurants in Chinatown than Chinese people. Gloria's own six-continent genetic blend was, perhaps, an extreme case, but not really unusual.
Gloria sat in the rear of the skimmer, smoking a jigli cigarette. Jigli was probably the strongest natural aphrodisiac in the Empire. She had discovered the herb on Mynjhino and still had a substantial hoard of an especially potent, refined version of it. This was her second jigli cigarette of the evening, and it was enough to get the shakes more or less under control. In addition to igniting a fire in her groin and producing an all-over tingling sensation in her skin, the jigli had a calming influence on her. At least, it allowed her to concentrate all her thoughts on one thing—sex—and avoid thinking of other things, like the smell of burned hair.
But the jigli was not enough. Not on this night. Not after this day.
Tonight, she needed something more.
The limo driver found the correct side street and halted in front of a subtle sign that marked the entrance to a subterranean establishment that went by the name of Club Twelve Twenty-Nine. In blue neon, the glowing outline of a clockface perpetually registered the time—12:29. Gloria got out of the limo and braved the brisk January air on her mostly bare body for the few seconds it took her to reach the door of the club.
She had been here a few times before during her years in Manhattan—usually with friends but also alone a couple of times. It was the kind of place that well-to-do young New Yorkers knew about from word of mouth, and could frequent in relative privacy and security. As Gloria entered, one of the massive bouncers, who could be counted on to keep media reps and imagers out, nodded to her and said, "Good evening, Ms. VanDeen. Nice to see you here again." Gloria gave the bouncer a smile, paused for a few moments to let her eyes adjust to the dim, half-light in the club, then headed for the bar.
Arnold was on duty behind the bar. Gloria knew him a little, the way one knew bartenders and waiters around town a little—not really as people, but as reliable fixtures. Arnold was a good-looking young man with somewhat dusky skin, very short, dark hair, and a worldly gleam in his eyes.
"Evening, Gloria," Arnold said. "You know, I had a feeling you might just show up tonight. Avatar of Joy, huh?"
"That's me," Gloria admitted. "How have you been, Arnold?"
"About the same," he said. "Altairian brandy still your drink?"
Gloria nodded, and Arnold went to fetch a bottle. The men on either side of her at the bar were staring at her breasts with obvious interest.
Arnold returned with the brandy bottle and a snifter. "Pour one for yourself, Arnold," Gloria told him. "On me."
"Don't mind if I do," he said. He produced another snifter and poured the amber liquid into it. They clinked glasses and sipped a little. He grinned at her. "Smooth," he said. It ought to have been; Altairian brandy was allegedly one of the best and certainly one of the most expensive liquors in the Empire.
At the back of the room, a few people were dancing to a small instrumental combo. They were playing twenty-seventh-century Syntho, which Gloria had always thought of as blues by and for computers. In the vast diversity of the modern Empire, and with fourteen or fifteen centuries worth of popular culture to draw on, there was no longer any dominant theme in music, art, or literature. Gloria's preference for twentieth-century blues was hardly unique, and around town one could find clubs that specialized in everything from nineteenth-century waltzes to the depressing, lugubrious ballads of the twenty-fourth century, to the minimalist syncopations favored by musicians of Gloria's own generation. Syntho was not at the top of her list of favorites, but she didn't mind it.
Gloria took another sip of the expensive brandy, then leaned forward to talk quietly to Arnold. "I think maybe lemon-lime tonight," she said.
Arnold nodded. "Whatever you want, Gloria," he said. "But have you tried the new wild cherry?"
Gloria shook her head. "I think I'll stick with the lemon-lime. I'm just a creature of habit, I guess."
"Pretty soon," Arnold said, giving her a conspiratorial wink, "you may want to change your habits."
"Oh?"
"Twenty-nine's divine," he said, "but Forty-eight'll be great."
"You have it?"
Arnold frowned. "Not yet," he said. "But the word is that it's being tested on half a dozen of the Core Colonies, and it ought to be available here in a few months. In the meantime, lemon-lime we got."
The bartender gestured like a stage magician and suddenly he was holding a small white lozenge between his thumb and forefinger. Gloria took his hand in hers, and when she pulled away again, Arnold's hand was empty.
It was called Orgastria-29. If jigli was the most powerful natural erotogenic substance in the Empire, Orgastria-29 was its bioengineered equivalent. Unlike jigli, which simply aroused erotic sensations, Orgastria-29 intensified them. It was one of a class of intensifiers, which flooded the brain with a carefully designed suite of neuroreceptors. Intensifiers could magnify, deepen, and enhance virtually any human emotional state. There were intensifiers such as "sobbers," which created an exaggerated sense of melancholy and produced a cathartic effect that was considered useful in some forms of psychological therapy and as a mood builder at certain concerts and poetry readings. And "laffers," which made almost anything seem hilarious. Orgastria-29—or Twenty-nine as it was known—intensified sexual sensations to the point of producing physical seizures at the moment of orgasm.
Twenty-nine, in fact, was so good at what it did that it was illegal. Orgastria-17 was a safe, legal, and very popular product. Twenty-nine could kill.
It didn't kill very often, but often enough for it to be registered as a controlled substance in the _Imperial Pharmacoepia._ Twenty-nine was generally taken in the form of flavored lozenges the user sucked on, getting small but steady doses of intensified sexual sensations. With her already-enhanced genetic equipment, and with the addition of jigli, Gloria was capable of sustained, continuing orgasmic episodes lasting more than an hour when she was under the influence of Twenty-nine. Most people got by with less than that, but a few wanted even more. They crunched down.
Crunching down simply meant grinding the Orgastria lozenge to powder between the molars. It produced an almost instant deluge of intensifying receptors that caused the brain to go into something similar to an epileptic seizure—a cerebral electrical storm. Most of the time the seizure was harmless and produced only an overload of ecstatic sensation that subsided after a few minutes. But sometimes, for some people, the seizure overwhelmed and shut down normal brain function. If a neutralizing agent could be administered quickly enough, brain function could be restored before permanent damage or death resulted. Clubs like Twelve Twenty-Nine generally kept a supply of the neutralizer on hand, thus avoiding the public embarrassment and legal difficulties of having people die on their premises. Nevertheless, people did die from Twenty-nine.
And so, people who wanted Twenty-nine had to get it from people like Arnold, in places like Club Twelve Twenty-Nine.
"Tell me about Forty-eight," Gloria said to Arnold.
"From what I've been told, it's like a whole new level," Arnold said. "Like crunching down on Twenty-nine, but it's sustained and you don't pass out. You just go to the top of the mountain and stay there an hour or so."
"Is it safe?"
Arnold shrugged. "I guess that's what they're testing. I don't think you'd want to crunch down on it, anyway. The zamies aren't idiots. They don't want to kill the clients, you know. By the time it's widely available, I'm sure it'll be as safe as Twenty-nine."
"Sounds good," Gloria said. "In the meantime, do you have a null-room free for tonight?"
"For an Avatar of Joy? Of course we do! I'll just extend our regrets to another party and set you up in Number Three."
"Thanks, Arnold. I really appreciate it."
"My pleasure, ma'am. I might even drop in on my break, just to make sure everything is going smoothly." Arnold smiled at her. Gloria smiled back.
Arnold held up a pad, and Gloria tapped it lightly with her left index finger, not bothering to read the numbers. The pad registered her fingerprint and did a first-order DNA scan, then charged the total to her account. The purchase of the Twenty-nine would be hidden by the expense of the brandy and the rental of the null-room.
"See you later, Arnold," Gloria said. She backed away from the bar and snaked her way through the crowd until she reached Null-Room Number Three. She tapped the entrance plate for another DNA scan, changed the Preferences indicator from Private to Open, then went into the room as the door slid open. She kicked off her shoes, slipped out of the scrap of band skirt, then dived forward.
She started to fall, but was quickly grabbed by the null field, and floated comfortably for several seconds until she arrived at the dead center of the room, equidistant from walls, floor, and ceiling. Gloria, who paid little attention to technical matters, was vaguely aware that this was not actual antigravity, but merely a convenient approximation of it created by an array of mass-repulsion units similar to the ones in skimmers.
In some ways, Gloria preferred null-rooms like this to the zero-gravity facilities that were available in some orbital installations. The absence of gravity was simply that, and you just bounced around like a Ping-Pong ball, which could be annoying. But in a null-room, it felt as if the air itself had thickened to the point where you could swim through it, which you couldn't do in zero G. The mass-repulsion fields pressed against you gently from all sides, like a ghostly massage, and merely floating there by yourself was sensuous and sexy.
But Gloria was not alone for long. People in the bar had seen her enter the room and noticed that she had designated it as Open. An engraved invitation could not have been more to the point. Half a dozen men and women quickly floated in after her, and Gloria slipped the lemon-lime lozenge into her mouth.
They came at her from every angle, every direction, and Gloria sighed in contentment. The tension and frustration and fear drained away from her, and she gave no further thought to Charles or DuBray or the Universal Church of the Spirit or to what had nearly happened to her on distant worlds. All that mattered was what was happening to her here and now in Harlem, in Club Twelve Twenty-Nine, in Null-Room Three. She felt the familiar jigli tingle and the sudden surge of intensity as she sucked on the lozenge, and surrendered herself to the rippling, roaring orgasmic ride that began within seconds of the first penetration of her golden, electric body.
It went on and on, and finally, when everything was just right, Gloria crunched down.
GLORIA WENT FLYING THROUGH THE AIR AND landed, very abruptly, ass first on the padding. She was stunned for a moment, not so much by the impact as by the fact that it had happened at all. Petra had actually thrown her.
Across the small gymnasium, Petra stood staring back at Gloria, a half smile on her face, almost as if she were embarrassed by what she had just done. Gloria shook her head and pushed herself up to her feet. "Got me," she said.
"First time."
Gloria nodded in acknowledgment. She had been trying to train Petra in the art of Qatsima for nearly six months, ever since their return from Sylvania. Three mornings each week, they reserved this small padded room in the Dexta Rec Center and Gloria attempted to indoctrinate Petra in the subtle, methodical, and wonderfully effective discipline of Qatsima. Part martial art, part acrobatics, part ballet, Qatsima had been developed centuries earlier by humans and the native species on the planet of Songchai. Gloria had taken it up after leaving Charles, principally as a way of keeping her body toned, but also as a means of self-defense; once away from Charles and his omnipresent security guards, she wanted to be able to take care of herself. She had developed a real flair for the art, and was now ranked as a Master—although she was well aware that the _true_ Qatsima Masters could have wiped the floor with her.
Yet this morning, even Petra had sent her flying. Of course, after last night at the Club Twelve Twenty-Nine, Gloria was short on sleep and hungover on wine, brandy, jigli, Orgastria-29, and a nightlong explosion of sex. She figured she was operating at no better than about 70 percent efficiency. Still, Petra had honestly bested her, and that had never happened before.
"You're getting good at this, kiddo," Gloria said. Actually, Petra was still pretty awful at it, and Gloria doubted that she would ever become truly skilled. But mastery of just a few moves, and the element of surprise, might save her in the future from what had happened to her on Sylvania.
Sylvania had changed their relationship as much as Mynjhino. Before Mynjhino, Petra had simply been her cute little assistant—flighty, smart-mouthed, and devoted. They had become best friends after Mynjhino, but Sylvania had somehow altered the balance between them. Petra, who was two years older than Gloria, no longer seemed like a little sister; on Sylvania, Petra had even asserted herself once and told Gloria to stop behaving like a self-pitying jerk. There was now a growing equality in their relationship that belied their relative positions in the Dexta hierarchy.
Gloria wearily assumed a crouch and spread her arms. "More?"
Petra shook her head. "That was one to quit on," she said. "Anyway, it'll give you something to think about the next few weeks while I'm away on New Cambridge."
"And give you time to gloat. Enjoy it while you can." Gloria picked up a towel and mopped away some of the sweat.
Petra picked up her own towel. "I'm going to practice on Pug," she said as she walked toward the door.
"Does he know that?" Gloria asked.
"Not yet," Petra cackled. "Hit the showers?"
Gloria shook her head. "I've got the room for another twenty minutes," she said. "And after the way you just humiliated me, I think I'm going to stay here a while and work on a few things. And maybe sweat off some recent excesses."
"Why are your excesses always so much more excessive than my excesses?" Petra wondered.
"Probably because you're smarter than I am. Go get your shower...before I decide to bounce you off the ceiling."
"Sore loser," Petra called over her shoulder as she left the room.
Gloria concentrated on some stretching exercises and leg raises, trying to ease some of the tightness she felt in her muscles. Twenty-nine tended to make the muscles spasm, and the day after a session could sometimes feel like the day after a triathlon. She had begun doing sit-ups when the door opened and a very large man entered the gym. He was at least a foot taller and a hundred pounds heavier than Gloria, with short, spiky blond hair and a thick neck. He removed the warm-ups he was wearing, leaving him in nothing but a breechclout and his rippling muscles.
Gloria got to her feet and said, "Excuse me, but I have the room for another fifteen minutes."
"Is that so?" the man asked. "Well, I think you're wrong. In fact, I know you are. This room is mine."
"If you'll just check the—"
"I don't have to check anything. The room is mine."
Gloria was becoming annoyed; more than that, a tickle of fear and suspicion began to play at the edges of her mind.
"You're Gloria VanDeen," the man said. "A Ten."
"That's right."
"I'm Erik Manko," he said. "Also a Ten."
"Level doesn't matter. I signed up for this gym, and—"
"Level matters a great deal," Manko contradicted. "In fact, in this case, it's the only thing that matters. As you know, Dexta has a time-honored method of resolving disputes between those of equal level."
_"What?"_ Gloria was aghast. "You can't be serious!"
"I'm perfectly serious," Manko said. "Let me show you how serious I am." With that, Manko calmly stepped toward her, balled his fist, and swung at her. Gloria saw it coming and just managed to duck, so that instead of breaking her jaw, his fist merely glanced off the side of her skull, stunning her and sending her spinning to the padded floor.
Gloria was dazed for a moment, but realized that she was in grave danger. As a coequal Ten, Manko could all but kill her if he wanted. The _Dexta Code_ forbade superiors from striking their inferiors, or subordinates striking their superiors, but it allowed those of equal level to work out their differences by physical force, if necessary. It was a fairly common occurrence at the lower levels, usually among Fifteens—just another aspect of the brutal Dexta routine. Gloria had been involved in such conflicts twice before. Once, as a Fifteen, another Fifteen had thought that he would teach her a lesson, and was, instead, taught a quick and painful Qatsima lesson by Gloria; then, on Mynjhino, she had literally kicked the ass of a traitorous fellow Thirteen. But a fight between Tens? Over whose gym it was? Gloria couldn't quite believe it, but had no time to ponder the meaning of it all. Manko was moving toward her, so Gloria instinctively assumed the Qatsima posture known as the Wounded Bird. Manko was in for a surprise...
Except that it was Gloria who was surprised. Manko countered with a Dancing Cobra that not only neutralized Gloria's Wounded Bird, but used it against her to get a grip on both of her ankles. _He knows Qatsima,_ Gloria thought as Manko whirled her around and threw her against one of the padded walls. She slammed into it with her back and shoulders, then sank to the floor.
Before she could recover her breath, Manko was on her again. She tried to claw at his eyes and knee his groin, but he was too fast and too big and simply overpowered her. Qatsima moves could defeat sheer size sometimes, but they had little hope of prevailing against both size _and_ Qatsima. Manko punched her hard in the belly, doubling her over, then picked her up and threw her against another wall. Then did it again.
Gloria began to lose track of what was happening. Somehow, she found herself stretched out across his knee, and he spanked her with the flat of his hand, the impact so hard that it sounded like the crack of a whip. Then he picked her up, whirled her around again, and threw her against another wall.
As she lay crumpled against the wall, her legs splayed, Manko approached her again, smiling slightly. "Don't worry," he said, "I'm not going to rape you. That would be against the _Code,_ wouldn't it?" Instead, he simply gave her a powerful, barefooted kick in her groin. And another.
"Why?" Gloria gasped. "Why are you doing this?"
Manko leaned over her and smiled in her face. "Administrator DuBray will be contacting you in a few days," he said, "after you've had a chance to heal. Don't disappoint him, Gloria. I'd hate to have to do this again."
He straightened up and started to turn to go, but stopped and looked back down at her. "No, that's not true," he said. "I'd enjoy doing it again." Then he left.
Gloria tried to get to her feet, but couldn't. The pain overwhelmed her, and she sank back to the floor and passed out. The person who had signed up for the gymnasium next found her there ten minutes later, bleeding and unconscious.
"NORMAN? IT'S GLORIA. I NEED TO SEE YOU." Gloria stood in the access booth at the Dexta VIP Transit and waited for Norman Mingus to respond to her audio signal. A few moments later, an overhead light turned green, and Gloria stepped through the Transit ring.
She stepped out into Idaho. The guard at the Transit booth nodded to her, quickly ran a detector over her body, then nodded again. "Welcome, Ms. VanDeen," he said. "Just go into the house and up to the third-floor observation deck. I believe you'll find Secretary Mingus there."
Gloria thanked him, then walked along the heated sidewalk, across a broad courtyard hip deep in snow, to the front porch of the home of Norman Mingus. Snow might have been scarce in New York these many centuries, but there was still plenty of it in the northern Rockies. The house looked like it might have been built by a nineteenth-century robber baron, with its gables and cornices, gingerbread and Victorian flourishes. In fact, Gloria knew, Mingus had built it barely ten years ago. With access via personal Transit—something available only to the very rich or very powerful—there weren't even any roads connecting the house with the rest of the world; it was completely isolated, two miles above sea level and twenty miles from the nearest town. As she stepped onto the porch, Gloria paused and looked around Mingus's aerie, seeing the snow-covered mountains and valleys as they must have appeared to the Crow and Blackfeet whose land this had been fifteen centuries ago.
An aide met Gloria inside the front door and led her up two flights of stairs to the third floor. He pointed toward the observation deck and departed. Gloria opened the outer door and stepped out onto an open-air platform with a sweeping view of the mountains marching off to the south. Mass-repulsion units preserved a bubble of warm air over the deck, but Norman Mingus sat in a comfortable-looking chair under a layer or two of thick blankets. He looked up at Gloria's arrival but didn't attempt to rise.
"Gloria," he said, "I'm sorry I haven't been available at the office the last few days. My doctors decided I needed a new pancreas, so they took one out of cold storage and plugged it in. I should be fine in a couple of—Spirit! What the hell happened to you?"
Gloria smiled lamely. "I had a little...confrontation...a couple of days ago," she said. "But it's nothing to worry about. I'll be fine." The doctors at Dexta had administered Quik-Knit to her three broken ribs and nanomeds that were already repairing her bruises and lacerations, cell by cell. In two or three days, no one would be able to tell that she had recently been beaten and bloodied. In the meantime, her entire body ached.
Mingus started to rise, but Gloria gestured him back down. "No, don't get up. In fact, I think I'll pull up a chair and join you. Looks pretty comfy." Gloria, in tight jeans and a tighter cashmere turtleneck, moved another chair next to Mingus's and sat down, sighing a little at the still-sharp pain in her ribs as she moved.
"Well?" Mingus asked, arching his eyebrows. Norman Mingus, at 131 years of age, had a face that was pink and almost unlined, with sharply etched aristocratic features and an unruly shock of thinning white hair. Gloria always thought he looked like a retired schoolteacher, or maybe a country parson. But there was nothing mild or passive in his blue-gray eyes; they accurately reflected forty years of experience in running a galactic empire.
Gloria scrunched around in the chair, trying to find a comfortable and pain-free position. "I ran into another Ten named Erik Manko," she said. "He pounded on me for a while."
Mingus nodded and sighed heavily. "So," he said, "you've finally run afoul of Cornell DuBray. I've been expecting it, but it honestly surprises me that he would resort to such means. Manko is his personal Hammer, you know."
"I didn't think there were any Hammers left in Dexta," Gloria said. "I mean, I've heard about them, but I thought they were..."
"History?" Mingus shook his head. "Not quite, although I did try. That was one of the reforms I introduced, years ago. Hammers used to be quite common around Dexta. Some Sector and Quadrant Administrators damn near had their own private armies. I tried to put an end to all that, but there are still a few individuals like Manko around. If you check his record, though, I'll wager you'll find that he was either promoted or demoted to a Ten the day before you encountered him."
"I did check," said Gloria. "You're right, he was demoted from a Nine. It seems he's been bouncing back and forth between Eight and Twelve for the last ten years."
"As required by DuBray. Tell me, what did you do to set all of this in motion?"
"I went to see DuBray about some routine business. Nothing more."
"That would be enough. I suppose I should have warned you. Cornell was steadfastly and vociferously opposed to the creation of OSI. Like a lot of the older hands around Dexta, he didn't see the need for it and resented the potential intrusion into his domain. But it surprises me that he would unleash Manko on you."
"Well," said Gloria, "there was a little more to it than that. DuBray insisted on having sex with me, then and there. Norman, he treated me like a Fifteen!"
"I see," said Mingus. "And of course, you resented that."
"Damn right, I did! I don't have to put up with that shit anymore. I'm a Ten now—"
"And he's a Four."
Gloria turned in her chair to look at Mingus. "That's what _he_ said. Are you telling me that it's okay? That he can get away with that?"
"Of course he can get away with that. Gloria, the man is a Four and has held his present position for over forty years! You've been a Ten for what, about a year?"
"About that," Gloria said sullenly. She didn't like the turn this conversation was taking.
"Gloria," Mingus said, a trace of impatience in his voice, "I bumped you from a Thirteen to a Ten. In case you didn't realize it, that was all but unprecedented in the history of Dexta. And I gave you powers and responsibilities at OSI that were far beyond those normally entrusted to a Ten. I suppose it's natural that all of that should have gone to your head. But let me remind you that as a Ten, you are just one of some six thousand Tens within Dexta. You are a midlevel bureaucrat, nothing more."
"Norman," Gloria said, "how many Tens come visit you here at your home?"
Mingus stared at her for a long moment, then smiled. "All right, then," he conceded, "perhaps you are something more than a mere Ten. In my eyes, at least—but not in DuBray's. And that's all that is truly relevant here. I'm sorry about what happened to you, Gloria. I regret it more than I can say, and I intend to have words about it with DuBray. Nevertheless, you cannot come running to me every time one of your superiors gives you a hard time. You will have to find your own way of coping with the Cornell DuBrays of this organization. I can't and won't intervene on your behalf."
Gloria felt appropriately abashed. "I understand, Norman," she said. "And I didn't really come here just to complain about the way DuBray treated me. There's something else. You know, don't you, that someone took a shot at me on Cartago?"
"Yes, I did hear about that. Very disturbing. Have there been developments I should know about?"
"I think so," Gloria said. "In fact, I need to ask you some questions, Norman. Questions that go back more than fifty years, to when you were the Quadrant 4 Administrator."
"Oh?"
"The weapon that was used in the attack on me on Cartago was originally part of a shipment of arms that was sent to Savoy in September of 3163, just before the Ch'gnth attack."
Mingus suddenly leaned back in his chair and seemed to focus his gaze on a distant mountaintop. He was silent for several moments, then breathed, "Savoy."
"And there's more. Another weapon from that same shipment was involved in one of the terrorist attacks in Quadrant 4 last week. It would seem that PAIN has somehow gotten its hands on weapons that were supposedly destroyed fifty-five years ago."
Mingus turned to look into her eyes. Then he looked once more at the distant snow-covered peaks and seemed lost in private thoughts or, perhaps, memories.
"Lewis and Clark returned from the Pacific through that pass," Mingus said at last, pointing to a notch in the mountains some thirty miles to the south. "Or, at any rate, Lewis did. They had split up by that point; then they joined up again down on the Missouri."
"Fascinating," said Gloria.
"I always wanted to follow the old Lewis and Clark Trail," Mingus said. "Never did, of course, and it's too late now. I'm too old, and most of it's gone, anyway. One more regret in a life that's full of them."
Mingus closed his eyes for a moment and sighed heavily. "You can't imagine what it was like, Gloria," he said. "Wartime, I mean. It was a truly desperate time. The Ch'gnth would have exterminated us if they could. In the end, we damn near exterminated them. But it didn't have to turn out that way. It could just as easily have gone against us. Believe me, I know—most of that damned war was fought in my Quadrant."
"I know. You're in all the history books. You'd have been remembered, Norman, even if you had never become Dexta Secretary."
"History books!" Mingus snorted. He took a sip of tea. "Let me tell you something, Gloria. If you live long enough to read about yourself in a history book, you'll never again trust anything you read in _any_ history book. Alexander, Washington, Napoleon, Churchill, Hazar—nowadays, I'm not even sure any of 'em really existed. Maybe it was _all_ lies."
"Did the history books lie about you?" Gloria asked him.
Mingus gave a short, bitter laugh. "Well, I'll just say that lies were told, and let it go at that."
He took another sip of tea and turned to look at Gloria. "I'm very concerned about this, Gloria. I'd always tended to dismiss PAIN as inconsequential idiots. Raving anarchist ideologues, the kind of people who used to throw bombs into the Tsar's carriage. But these recent attacks are clearly well coordinated, and now it would seem that you were one of their targets."
"A symbolic one, at best." Gloria shrugged.
"Those are usually the most important ones," Mingus pointed out. "You, my dear, are perhaps the most visible symbol of Dexta and the Empire these days. In the warped view of the anarchists, killing you would represent a considerable triumph. I am going to order additional security for you, and you will accept it without complaint. Is that clear?"
"Yessir," Gloria answered obediently. Mingus stared at her for several seconds, as if trying to assure himself that she had meant what she said.
"The question I really want to answer, though," Gloria told him, "is how PAIN came by those weapons from the Savoy shipment. They seem to have enough of them to carry out several nearly simultaneous, widely separated attacks. What can you tell me about that shipment, Norman?"
Mingus shook his head wearily. "It was a long time ago," he said.
"But Savoy was a pretty significant event," Gloria pointed out.
"They were _all_ significant events! You simply have no idea, Gloria—none! In those days, the very fate of the human race hung in the balance. We could have lost it all—the Empire, even our very existence as a species. In those weeks leading up to the onset of hostilities, I found myself making decisions on a daily basis that might affect the course of events for the next millennium. That final shipment to Savoy...it was just one more. Spirit knows what finally became of it."
"I see," said Gloria. "And you don't remember any details about it?"
"Not really," he said.
"Well, I'm sending an OSI team to New Cambridge to dig around in the original records. Maybe if we can find out what really happened to that shipment, we can figure out how PAIN got their hands on it."
Mingus looked back toward the distant peaks. "Perhaps so," he said. "In the meantime, be very careful, Gloria."
Gloria sensed that she was being dismissed. She still had many more questions, but decided that this was not the time to ask them. Still, there was one thing more that she needed to say.
"Norman? I think you should know about this. Charles wants me back. He wants me to be Empress, and he's offered me real and meaningful power."
Mingus raised an eyebrow. "Has he, now?"
"He says we should pool our resources instead of competing. He says that together, we could be the best leaders the Empire has ever had."
"He might just be right about that," Mingus said, nodding thoughtfully. "On the other hand, between the two of you, the Empire might just find itself plunged into an all-out civil war. What did you tell him?"
"I said I needed time to think about it. There's a lot to consider. I never wanted to be Empress and felt lucky to have divorced him when I did. But to have real power...I just don't know. I can't deny that it has an appeal."
"Of course it does. Somebody once said that power is the ultimate aphrodisiac. I won't say that power is better than sex, but it's at least as good. Seems that way at my age, at any rate."
"But I have some power at Dexta, too," Gloria said. "And I've _earned_ that. In time, maybe I'll earn more."
"How much more?" Mingus gave her a long, probing stare. "Just what are your ambitions, Gloria? What do you hope to achieve at Dexta?"
"Norman," she said, "I intend to be your successor. I intend to run Dexta someday."
Mingus took a moment or two to react, then shook his head. " _My_ successor? My successor's successor, perhaps...but mine? Gloria, the youngest Secretary in Dexta's history was, I believe, fifty-seven. You are twenty-five, and I am a hundred and thirty-one. Just how long do you expect me to live, anyway?"
Gloria smiled at him. "Forever, I hope." She leaned over and gave him an affectionate peck on his cheek. "People live to be nearly two hundred, you know."
"And I have no intention of becoming one of them. After a hundred and fifty, it's just not worth it. A hundred and thirty is no day at the beach, either, I can tell you. I will say this much, though. I'll never resign my position. I've been employed by Dexta for one hundred and seven years; it's the only paycheck I've ever drawn. I'd die without Dexta, so I might just as well die within Dexta."
"I'd take it as a personal favor," Gloria said, "if you put it off for another twenty or thirty years. That might be enough to give me a shot at being the next Secretary."
"It might, at that," Mingus conceded. "But you must realize that if I drop dead tomorrow—and my doctors refuse to promise me that I won't, new pancreas or no—my obvious successor would be Cornell DuBray. He would have the Dexta vote locked up, and he has more than enough influence with Parliament. He'd probably get the Emperor's vote as well, if it came to that."
Dexta Secretaries were chosen democratically, but the electorate was limited and elite. Parliament, senior Dexta personnel (Sevens and above), and the Emperor each had one vote. Two votes were necessary for a candidate to achieve office, and the appointment was effectively for life, since all three votes were necessary for removal from office. Mingus had been nominated by Darius IV in 3176, and had been confirmed with three votes. Charles might have liked to remove him and install a younger, more pliable Secretary, and there were those in Parliament who thought that he had been in power far too long; but his support among upper-level Dexta people was unquestioned. If both the Emperor and Parliament turned against Mingus, his support within Dexta would probably erode in time, but it would have taken a major scandal to dump him. Practically speaking, the office was his for as long as he wanted it.
"Gloria," Mingus said, smiling, "it seems you have a rather extraordinary choice to make. You can achieve immediate and certain, if limited, power by becoming Empress. Or you can spend decades working to earn a deeper and more meaningful power by attempting to become Dexta Secretary. I doubt that anyone in the history of the Empire—or the human race, for that matter—has ever faced such a choice, or such an opportunity. Choose wisely, my dear."
THE NEXT MORNING GLORIA MET WITH SOME of her senior staff in a small conference room. Gathered around the table, sharing coffee and bagels, were OSI Administrator Grant Enright, Level X; Deputy Administrator Jillian Clymer, Level XI; OSI Internal Security Coordinator Arkady Volkonski, Level XII; and Elaine Murakami, Level XIV, subbing for Petra as Gloria's assistant and gatekeeper.
Enright was a conventionally handsome, fortyish Lion who had been Gloria's boss back in Sector 8. He was happily married—a rarity in Dexta, where marriage by lower-level personnel was actively discouraged—and enjoyed the distinction of being one of the few human males who had ever said no to Gloria. He had little patience with the sexual stratagems so often employed in Dexta and had become Gloria's friend rather than her lover. Enright was efficient, friendly, and soft-spoken, and seemed to have been born to be an administrator. He kept OSI functioning with brisk efficiency during Gloria's frequent absences.
Yet for all his abilities, Gloria had lately come to realize that Enright lacked the inner creative spark that distinguished true leaders. Given everything that had happened or almost happened to her in the past year, Gloria had inevitably been driven to give some thought to a possible successor for herself at OSI, and she doubted that Enright was the best person for the job. A better candidate, it seemed to her, was Jillian Clymer, who had been recruited for the Sylvania mission and had then accepted Gloria's invitation to join OSI on a permanent basis.
Jillian was a curvy, apple-cheeked blond from a small agricultural world in Sector 2, whose radiant smile, enthusiasm, and positive attitude had made her a welcome addition to the OSI staff. She had inherited a modest fortune and, like Gloria, had married young and divorced young before joining Dexta. In the Dexta bestiary she would have been classified as a Lion, but, like most women in OSI, she had begun to take on some Tigerish traits under Gloria's influence. Today, she wore a standard, Dexta-gray skirt and a matching V-necked sweater with, very obviously, nothing under it. Arkady Volkonski had apparently taken note.
Volkonski, a dark, glowering Cossack—distinguished by his ingrained sense of irony and a single, unpunctuated eyebrow—professed unlimited devotion to Gloria. But he was no Lap Dog; he was more of a Rottweiler. He took any assault on Gloria—from outside Dexta or from within—as a personal affront.
Enright led off the meeting by bringing Gloria up to speed on various administrative matters. Elaine Murakami nibbled at the edges of a bagel while taking notes on her pad. She had worked for Gloria back in Sector 8, before the creation of OSI. A thin, attractive, would-be Tiger, she was energetic and loyal, but had somehow failed to impress Gloria with any deeper qualities. And Gloria found her hero-worship frankly annoying. She was not comfortable with the notion of being anyone's role model, no matter what the Spiritists said about her.
When Enright had concluded his presentation, Gloria turned to Jill Clymer. "Where do we stand on the double-flagging in Sector 19?" she asked.
"Nothing new to report," Jill said. "As you know, I sent a team from Finance out to Staghorn last week, but it's five-hundred-and-some-odd light-years. Their Cruiser should have arrived yesterday, which means we could get an initial report via messenger in a few days." Messengers were reusable automated spacecraft that traveled at a rate of about 150 light-years per day; they were the standard means of disseminating information within the Empire. For more urgent communications, courier spacecraft could do two hundred light-years in a day, which meant that Earth was no more than five days away from the frontiers of the Empire. However, couriers were expensive, since they carried no fuel for deceleration and could, therefore, be used only once.
"And the initial report was filed by the Imperial Secretary on Staghorn?" Enright asked.
Jill nodded. "She seemed to regard it as something of a hot potato. One of her Finance people came to her with what looked like some suspicious data, and she took one look and sent it off to us. Seems she didn't want to have to deal with some of the big names involved."
"Big names?" Gloria asked.
"Big locally, at any rate," Jill explained. "The evidence, such as it is, points toward an outfit called Wendover Freight and Storage, which is based on Staghorn and is the major local transport company in the Sector. It also does business in Sectors 20 and 21."
"Big Twelve connections?" Gloria wondered.
"None, as far as we can tell," Jill said. "Of course, it's not always easy to sort out corporate connections. I've got someone checking out their history; Wendover has been in business nearly a century, so you would expect at least some Big Twelve affiliations. But nothing has turned up yet."
"That in itself seems a bit odd," Gloria opined.
Jill shrugged. "There are a few genuine independents left, and Wendover may be one of them. Still, if they are engaged in a large-scale double-flagging operation, it would make sense for them to have some ties to one or more of the Big Twelve."
"Or not," Gloria suggested. "Maybe the lack of a profitable connection to the big boys is what's driving them into the double-flagging scam. Just how good is this evidence? And what about the Dexta office on Staghorn? You said the ImpSec was nervous about dealing with this. Could there be Dexta involvement?"
"Your guess is as good as mine," Jill said. "But to tell you the truth, it wouldn't surprise me. A good double-flagging scam takes inside help. I've checked out a dozen or so examples of this kind of thing over the past five or six centuries, and every one of them involved someone in Dexta. And I'll bet the ImpSec on Staghorn thinks so, too—which is why she sent this to us instead of the Comptroller. Gloria? Are we going to get any help on this from Quadrant?"
"No," Gloria answered flatly. "Probably just the reverse."
"I see," Jill said. "DuBray's determined to be an asshole?"
"It's what he does best," Gloria said.
"Speaking of DuBray," Volkonski put in, "I still say you should let me deal with this Manko character."
Gloria reluctantly shook her head. "You know I can't do that, Arkady," she told him.
"Then just look the other way and let me handle it in some unofficial, unobtrusive way. Like maybe kneecapping the son of a bitch in some dark alley."
Gloria smiled, then reached out and ran her index finger along the underside of Volkonski's chin. "You're such a sweet, sensitive man, Arkady," she said. "But I can't let you risk your career over this. It's my problem; I'll deal with it myself."
"Then promote me to a Ten," urged Volkonski, "and let me be _your_ Hammer."
"Not possible," Gloria reminded him. "Anyway, I'm not sure you could take him in a fair fight."
"Who said anything about a fair fight?" he asked. "You forget, I'm a Bug."
But it had to be Gloria's own fight, she knew, fair or not. When she was a Fifteen, she had endured worse than anything DuBray had done so far. If she had survived that, she could survive this. And, she asked herself, if she couldn't find a way to handle DuBray, did she really deserve to be Dexta Secretary? Dealing with reptiles like DuBray, after all, was part of the job.
"And speaking of Bugs," said Enright, "what is Internal Security doing about PAIN?"
The previous day, Internal Security had convened an upper-level staff conference to address the terrorist problem. Despite his relatively low rank in the Dexta hierarchy, Volkonski had been invited to the meeting in recognition of his role at OSI and the fact that Gloria had been targeted.
"There are things that I can't discuss with you," Volkonski said, looking a little guilty about it.
"We understand that, Arkady," Gloria said. Internal Security was notoriously stingy with information, even within the ranks of Dexta. "Just tell us what you can."
"Well, as you might imagine, the three attacks on Dexta personnel have caused something of a stir in IntSec. The assumption is that the PAIN faction responsible is based in Quadrant 4, since two of the attacks were there—although Cartago is in Quad 2. Anyway, measures are being taken. One might usefully surmise that said measures would include rousting every known or suspected radical in the Quadrant and shaking them until their teeth rattle. Perhaps something worthwhile will emerge."
"And that's it?" Jill asked.
"What would you have us do, Jillian?" Volkonski asked sadly. "There are a limited number of us, after all. Even working night and day, there are only so many citizens whose civil rights we can violate. Anyway, the real problem is that PAIN is, paradoxically, an organization of anarchists. In other words, it's no 'organization' at all—just a voluntary, free-form convergence of like-minded morons. Supposedly, there are only Indians, no chiefs."
"Still," Enright insisted, " _somebody_ must be calling the shots. Three nearly simultaneous attacks spread across two Quadrants couldn't be a coincidence."
Volkonski scowled and thought for a moment. "One might presume," he went on, "that an organization such as Internal Security, when confronted with a threat such as PAIN, might attempt to infiltrate the opposition and glean information from the inside. Such information might even now be winging its way toward Earth. If such were the case, then an organization such as Internal Security might be weighing options and preparing possible responses in anticipation of the receipt of such information."
Jillian gave him a cool, level stare. "You don't have a clue, do you?"
Volkonski shook his head.
That afternoon, as Gloria was trying to catch a few surreptitious winks at her desk, a pinging sounded and the face of Elaine Murakami appeared on her console. "Gloria? Can you take a meeting with Eli Opatnu? He's the—"
"I know who he is," Gloria said. She had never met him, but he was well-known within Dexta and was considered to be something of a rising star. Not yet forty, he was already a Level VII, the Administrator of Sector 19. "When does he want to see me?"
"Right now," said Elaine. "He's here."
"Okay," Gloria said, "give me a minute, then send him in." Gloria got to her feet, tried to rub some of the sleep out of her eyes, then gave her head a good shake to signal her long, flowing mane to restore itself to Dura-styled order. She took a quick glance in the closet mirror and was reasonably satisfied with what she saw. Her pale yellow minidress was relatively modest—perhaps too modest, it occurred to her when she thought of the images she had seen of Eli Opatnu. She pressed a hidden contact in a seam and the smart fabric of the dress retracted. She pressed another contact, adjusted the opacity of the fabric to 40 percent, then declared herself ready to meet Opatnu.
The Sector 19 Administrator entered her office a moment later. Elaine closed the door behind him and Gloria extended her hand. "Hi," she said, smiling broadly, "I'm Gloria VanDeen. Very pleased to meet you, Mr. Opatnu."
"It's Eli," he said, taking her hand. He grasped it firmly in his own and held it while he slowly surveyed every inch of her face and body—and Gloria did the same with his. Opatnu was the product of the same sort of genetic stewpot that had created Gloria, and his face seemed to bear traces of the Andes, Himalayas, and Alps. He had longish, jet-black hair with a hint of a wave in it, a strong, European nose, vividly green eyes, and skin that reminded Gloria of the color of an old saddle. He was tall and lean, but there was nothing fragile about him. His hand in hers was warm and forthright, with long, almost delicate fingers.
Gloria tried to catch her breath. _Why don't we skip the preliminaries,_ she wondered, _and just start screwing here and now?_ From the look in his eyes, she figured that Opatnu was probably thinking the same thing.
They finally released their grip on one another. "I'm glad you dropped by," Gloria said. "I was planning to pay you a visit in a couple of days. Please, have a seat."
She sat next to him on the couch, and they spent a few more seconds silently smiling at each other. She could almost see the waves of heat rising between them. Gloria was never shy or reticent where sex was concerned, but she couldn't recall the last time she had been so thoroughly overwhelmed by a man's sexual presence. Charles, maybe, long ago, when she was young and easily impressed. But not many since. Eli Opatnu was downright breathtaking.
"Thank you for seeing me on such short notice," Opatnu said at last.
"My pleasure," Gloria said.
"Mine too," he said, breaking into a grin. His teeth were so white that it almost hurt to look at them.
Gloria started to reach for him, almost reflexively, but managed to divert her motion in another direction, and wound up standing up again, even though she had just sat down.
"Uh...can I get you something to drink...Eli?"
"No thank you...Gloria. I'm fine."
"Yes, I can see that. I mean, uh, it's a pleasure to finally meet you. I've heard a lot of good things about the work you've done in Sector 19."
"I have a full squad of creative publicists at work night and day," he explained. "Oddly enough, I don't think I've ever heard _anything_ about you."
"I'm the shy, retiring type," Gloria said. "And may I ask, what brings you here today?"
"I think you know," Opatnu said.
Gloria nodded, then sat down next to him again.
"You've started an investigation in my Sector," he said. "I hoped you could give me an explanation."
"Perhaps," Gloria said. "But first, I'd appreciate it if you could tell me how you heard about the investigation."
"Oh," he said vaguely, "I have my sources."
"I'm sure you do," Gloria agreed. "And I'd like to know who those sources are. An OSI investigation is supposed to be secret, at least at this point in the process. Sector Administration has not yet been officially notified."
"I understand that, Gloria," Opatnu said. "However, I'm sure you must understand that I can't reveal my sources. Let's just say that I keep an ear to the ground."
"I don't want to get off on the wrong foot with you...Eli," Gloria said, meaning every word. "But I'm very concerned about this lapse in OSI security. By any chance, did you hear something about this from anyone in Quadrant?"
Opatnu shrugged. "If I did—and I'm not confirming or denying anything—I certainly wouldn't be at liberty to tell you. Let it go, Gloria. Pursuing this isn't going to get either one of us anything."
"It's not something I can ignore, Eli. If someone from this office—or elsewhere in Dexta—is leaking restricted information, I need to know about it. I cannot have OSI operations compromised, or my people endangered, by lax security and loose lips."
"I seriously doubt that anyone is being endangered," Opatnu said. He sat up a little straighter on the couch and his gaze was no longer as open and friendly as it had been. Neither was Gloria's.
"I have a team on Staghorn," she said. "If someone has already sent word of their mission via messenger or courier, it's possible that they could be in some jeopardy. This is not simply some bureaucratic turf battle, Eli. I need to know how you found out about the investigation."
"I can't help you," Opatnu said. He got to his feet and she got to hers.
"Then I can't help you, either. You'll be informed about the investigation in due course, through proper channels. And if you or any of your people in Sector get in the way, the OSI will come down on you so hard, you'll wonder what hit you."
Opatnu nodded. "All right," he said, "if that's the way you want to play it." He took a step toward the door, then pivoted sharply, grabbed Gloria by her shoulders, pulled her to him, and gave her a long, smoldering kiss that she found herself returning with full intensity. It was at least a minute before they broke.
"That doesn't make any difference," Gloria told him.
"I know," he said, and left.
But it did.
THE COLONY ON NEW CAMBRIDGE, 362 LIGHT-YEARS from Earth, was founded in the year 2367 by a party of young emigrants who had recently graduated from Harvard and the Massachusetts Institute of Technology. Wishing to make it clear that the Cambridge they were honoring was the one in New, rather than Old, England, they assigned the major settlements on the planet's two continents place names that came from the ancient American city they had left. Thus, on the west coast of the western continent stood the great city of Brattle, complemented on the east coast of the eastern continent by the metropolis of Kendall. Between the two lay the planet's greatest city, Central. It was situated on the east coast of the western continent, facing the Dardanelles-like, thirty-mile-wide strait that separated the two landmasses.
In the eight-and-a-half centuries since the colony's founding, the population of New Cambridge had swollen to over 10 billion, and it had become the most important and prosperous world in all of Quadrant 4. Indeed, in the entire Empire, only Earth and two or three other planets could rival it. Pug Ellison's ancestors had been among the founders, and his family had done its part in promoting the colony's growth. They had profited accordingly, as Petra fully realized for the first time when the limo skimmer brought them to the gates of the Ellison Compound north of Central, high on a shelf of the cliff overlooking the straits.
"Your front yard is bigger than Weehawken," Petra noted.
"Yes, but we don't have a backyard at all, thanks to the cliff face."
"No backyard? Driver, take me back to the port! I'll be damned if I'm going to stay with some lowlife no-account who doesn't even have a backyard."
"Would it help if I told you that I have a bed that's also bigger than Weehawken?"
"Suitable for team sports, no doubt?"
"I prefer one-on-one, myself. But if you'd like, we could call in the staff and choose up sides."
"Hmm," Petra hummed. "We'll save that for later, I think...in case I get bored in this cheap dump of yours. Spirit, Pug, just how rich _are_ you, anyhow?"
"So rich," he said, "that numbers really have no meaning. Seriously, I couldn't tell you the family's net worth within the nearest trillion crowns."
They debarked from the limo at a side entrance to the immense edifice that Pug called home. The building reminded Petra of an ancient European cathedral. It appeared to have been hacked out of the same rocks as the cliff face that soared behind it, looming a thousand feet above in the crisp, blue sky. She had never seen a house with flying buttresses.
The inside was cavernous and as intimidating as the exterior. The entrance anteroom—Petra thought of it as a mudroom—was larger than the apartment she had grown up in. Pug led her onward through columned corridors and nameless rooms hung with vast, ornate tapestries. Their footsteps echoed on the polished marble floor.
"Where are the gargoyles?" Petra wondered. "There ought to be gargoyles."
"You mean my parents?" Pug asked with a smirk. "Right this way, I think."
Sure enough, the next room they entered contained an immense hearth with a blazing fire, complex chandeliers hanging from a high, vaulted ceiling, plush furniture of mahogany and gold satin, and sitting on it, Pug's parents. They got to their feet and greeted their son with smiles that seemed cool but genuine. Pug didn't quite hug his mother, but gave her an affectionate kiss on her cheek, then turned to his father and shook hands as if closing a business agreement.
"Mother, Father," he said, "you're both looking well. It's wonderful to see you again."
"It's been more than a year, dear," his mother said. "Far too long. And who is this delightful creature?"
"This," Pug said proudly, "is Petra."
Petra stepped forward and extended her hand to Mrs. Ellison, who offered her own in return. It was cool and limp. Mr. Ellison's grip was strong and confident, and he gazed at Petra with evident appreciation. Pug's parents, she knew, were both in their eighties but, thanks to antigerontologicals, looked half that. His mother was slim and fair-haired, with frosty blue eyes and an arctic smile. His father had the same friendly, open face as his son, but with a hint of gray in his hair and crinkly, skeptical eyes.
"So pleased to meet you at last, dear," said Mrs. Ellison. "Palmer has mentioned you, of course, but it's so difficult to truly know someone based on sketchy descriptions."
"But accurate, it would seem," added Mr. Ellison. "I recall words such as 'beautiful' and 'wonderful' and 'charming.' "
Petra found herself on the verge of a blush. She had dressed for this grand meeting in a modest but slightly saucy blue dress, with a neckline deep enough to have attracted Mr. Ellison's attention. This was the first time she had ever really gone through this particular ritual—Meeting Your Man's Family—and she wasn't entirely sure how to approach the event. The fact that it was taking place in a room the size of a Transit station complicated matters; what should have been an intimate gathering seemed more like a state occasion. She felt like an ambassador presenting her credentials to some lofty monarchs.
"It's wonderful to meet you both," Petra said, smiling. She looked around her. "Nice little place you have here."
"It serves," Mr. Ellison said modestly. "But any home can always stand to be brightened by the presence of a beautiful and charming young woman."
"Indeed," said Mrs. Ellison. She took Petra by the hand and led her to a divan. "Now, dear, sit down and tell us all about yourself."
"I WAS BORN IN NEW JERSEY, AND I'VE WORKED for Dexta two-and-a-half years." Petra slapped herself in the forehead and flopped over backward onto the bed-the-size-of-Weehawken. "Spirit! I sounded like I was applying for a job. I sounded like an idiot!"
"Most everyone does when they first meet my parents," Pug said. "Ordinary people tend to get tongue-tied in the presence of overblown opulence. But you'll get over it."
"Will they? I mean, first impressions are supposed to be important, aren't they? By now, your parents have probably concluded that their son is infatuated with some brainless peasant girl."
Pug smiled. "They'd have thought that no matter what you said. Relax, they'll get over it and so will you. They're just people, Petra."
"Yes," Petra protested, "but I can tell that they were expecting...more, somehow. I mean, you told them I was beautiful. _Gloria's_ beautiful! I'm just...cute. Cute and little."
"You look beautiful to me," Pug said, staring at her naked body. "And as for your size, I think we fit just fine." Pug crawled on top of her and started to demonstrate, but Petra twisted over and tried to crawl away from him.
"Race you to the other side," she challenged.
Pug grabbed her leg and held her in place. "Too dangerous," he said. "You might get lost under the sheets. Even worse, you might run into some of my old girlfriends under there."
That got Petra's attention. She reached back and grabbed Pug by the nose. "Old girlfriends?" she demanded. "You told me they were all dead, or in convents."
"Well, they are, with one or two exceptions... _oww_!"
"And what about those exceptions, huh?"
"Well," Pug said, trying to extract his nose from Petra's grip, "you might meet some of them at the party next week. I wouldn't worry about it if I were you. I mean, you can see how big this bed is... _yowww_!"
"That's a Qatsima grip Gloria taught me. Would you like to see some more?"
"Not just now, thank you." Pug managed to free himself.
Petra stretched out on her back and stared upward, trying to make out the details of some preposterous mural on the ceiling. Pug lay next to her and amused himself by playing with her small, round breasts. He toyed with her nipples until they were fully erect, then leaned over and kissed them. Petra squirmed for a bit, then sighed contentedly.
"Are those billygoats or satyrs up there?"
"What kind of parents would put satyrs on their son's ceiling? They're goats, and those are cows, and pigs, and chickens...the Ellisons started out here as farmers, you see. That section of the mural commemorates our agricultural roots. Over there, you can see the industrial era, and off in that corner you see commerce, science, and philanthropy."
"And what kind of parents would put the family history on their son's ceiling?" Petra wondered.
Pug shrugged. "Ambitious ones?"
"How do they feel about their only son being at Dexta?"
"They aren't thrilled about it," Pug conceded.
Petra propped herself up on one elbow and looked into Pug's face. "And how do they feel about your being my assistant?"
"That," Pug said with a rueful nod, "does bother them a bit. Understand, it's not simply that _you_ outrank me. It's that _anyone_ outranks me. They think I ought to have Mingus's job. Well, in a year or two, anyway."
"Can't have it," Petra told him. "That job's reserved for Gloria. And when she gets it, I'll be her gatekeeper...which will make _me_ the most powerful woman in the Empire!"
"Be sure to mention that to my parents. _That_ would impress them."
"I'll make a note of it." She put her arms around his neck and pulled him on top of her. "Now, about those old girlfriends. Did any of them ever do _this_?"
"Is that some kind of Qatsima move?" Pug managed to grunt.
"Nope. That's a _Petra_ move."
"Better yet," Pug gasped.
GLORIA MARCHED THROUGH THE OUTER OFFICE of Quadrant 4 Administrator Cornell DuBray and into his inner sanctum, feeling a mixture of resentment and trepidation. He had summoned her and she had come, but she still had no clear idea of how she would handle the situation. Her indecision was even reflected in her clothing—a relatively modest dark miniskirt, and a light blue, nearly transparent blouse which she had finally decided to leave unbuttoned to the waist. She was determined not to be intimidated by DuBray; on the other hand, Eric Manko intimidated the hell out of her.
She found DuBray waiting for her in his inner office, flanked by the three other Quadrant Administrators. They were standing next to one another, like a firing squad. Gloria stared at them in silence for several moments. No one offered her a chair.
One of them, Manton Grigsby from Quadrant 2, Gloria already knew slightly from her days in Sector 8. He was a small, dapper, efficient man, and had held his present position for just six years, which made him the junior Quad Admin. Next to him, on DuBray's left, stood Mustafa Algeciras, a swarthy, somewhat portly man from a colony world in Quadrant 3. He was the highest-ranking nonterrestrial in Dexta, making him highly visible and politically indispensable.
Standing to DuBray's right was Elsinore Chandra, a woman Gloria had admired from afar for years. She had been the Quadrant 1 Administrator for nearly as long as Gloria had been alive, and was a certified Dexta legend. Chandra was over a hundred now, but still looked slinky and seductive, with a long woven braid of dark hair thrown casually over her left shoulder and descending nearly to the floor. Her features were sharp and distinct, with large brown eyes that appraised Gloria with professional interest. Chandra had been the Gloria VanDeen of her day, Dexta's most prominent Tiger of the mid-thirty-second century. Whenever Gloria found herself wondering if it was really possible to succeed as a Dexta Tiger, she would think of Chandra and know that it was.
"Thank you for coming," DuBray said with a perfunctory but oily smile. "Mr. Grigsby, Mr. Algeciras, and Ms. Chandra and I thought it might be useful for you to meet with us and get a few things straight. We probably should have done this sooner, but we wanted to see how you and your Office of Strategic Intervention performed. Frankly, some of us didn't expect you—or it—to last very long."
"I'm still here," Gloria said. She nodded toward Grigsby. "It's nice to see you again, Mr. Grigsby."
"And you, Ms. VanDeen," Grigsby said, smiling a little. Gloria stepped toward him and shook his hand.
Then she offered her hand to Algeciras, who took it in both of his own hands, kissed it, and looked into her eyes. "A pleasure to meet you at last, Ms. VanDeen. I look forward to getting to know you much better, but for now, it seems we must tend to business." He released her hand as if letting a captive bird fly away.
She didn't bother to greet DuBray, but turned to Chandra and offered her a warm smile as they clasped hands. "Ms. Chandra," she said, "it's an honor to meet you. I've been an admirer of yours for a long time."
"And I have taken note of you, as well, Ms. VanDeen. You play the game rather nicely, I think. You are off to an excellent start at Dexta. But it would be a shame to see such a promising career derailed."
"Derailed?"
"It's not really your fault," Chandra said. "I blame Norman Mingus entirely. I told that old goat what a monumentally bad idea this Office of Strategic Intervention of his was. But he hasn't listened to me since I kicked him out of my bedroom, twenty years ago. I gather that you aren't fucking him, though, are you? Is that your choice, or his?"
"I don't think that's really any of your business," Gloria replied.
"No," Chandra said, "I suppose it's not. No more than it's any of my business that Cornell sicced his pet psychopath on you for not putting out." She gave DuBray a not-very-friendly glance.
"I seem to recall a time," DuBray said, smirking at Chandra, "when you had a virtual platoon of handsome young Hammers at your disposal, Elsie."
"Maybe I did," Chandra sniffed, "but at least I never had to have anyone beaten up to get him to go to bed with me." She looked back at Gloria. "I admire your grit, Ms. VanDeen, if not your good sense. You must realize that if you intend to remain in Dexta, the day will eventually come when you find yourself having sex with Cornell DuBray. It's not really as awful as you might imagine. I lived through it, dear, and so will you."
"Thank you, Elsie, for that ringing endorsement. But leaving personal matters to one side for the moment, the purpose of this meeting, Ms. VanDeen, is to acquaint you with some realities. The four of us are presenting what you might call a United Front against the Office of Strategic Intervention, and we wanted you—and Norman—to understand that we mean business."
"It's true, you see," said Algeciras, spreading his arms a little, as if in apology. "The OSI has no logical place in the structure of Dexta. Its sudden growth is like that of a malignancy in an otherwise healthy body. And like a malignancy, it must be excised."
"Putting it a little less colorfully," Grigsby said, "we see OSI as counterproductive and contrary to the best interests of Dexta. Its charter infringes on the inherent powers of Quadrant, and even Sector, Administrators. The very existence of OSI implies the incompetence of the normal bureaucracy. Dexta is perfectly capable of functioning without the aid of highly public 'strategic interventions.' "
"What he means, dear," said Chandra, "is that you make us look bad. The public loves it, of course, but it's bad for morale among Dexta people. If you're the good guy in these little dramas, that makes the rest of Dexta look like bad guys. And we're not"—she paused and eyed DuBray for a moment, then continued—"or most of us aren't, at any rate."
"We've discussed all of this with Norman," DuBray said, "but he refuses to accept reality. I wonder sometimes if he's really up to the demands of the job anymore. In any case, he intends to continue his little experiment, regardless of the consequences. Of course, the consequences for Norman won't be nearly as painful as they are for you."
Grigsby frowned and said, "That's not really necessary, DuBray." He turned to Gloria and said, "While I may not agree with my colleague's methods, I am in complete agreement concerning his goals. We all are. Norman Mingus doesn't run Dexta, Ms. VanDeen— _we_ do. You should keep that in mind."
"Whatever the OSI Charter may say," Algeciras told her, "as a practical matter, you will need our cooperation in order to function. And that you will not get."
"What's more," said Chandra, "most of the Sector Administrators support our position. Wherever OSI attempts to operate, you will find not only no cooperation, but determined opposition. I realize that the OSI Charter gives you certain peremptory powers, and Norman will certainly give you as much leeway as the _Dexta Code_ allows. Nevertheless, within Dexta the OSI will be treated as a pariah. Think about what that would mean—for your office, and for you personally, dear."
Gloria had stood there listening in silence, all the while feeling a rising tide of emotion within her. Most of it was anger, along with outrage and defiance. But there was fear there, too. The most powerful people in Dexta had just declared war on her and OSI.
She took a deep breath. "Norman Mingus created the OSI and appointed me to lead it. If he should decide that he made a mistake and wants to remove me or dissolve the office, that would be entirely up to him. Not you—him. Until that happens, I intend to do my job to the best of my ability, come what may. If you expect me to quit, you'd better think again."
DuBray shook his head. "No, we don't expect you to quit," he said. "You have become very famous and popular with the masses, and we have no intention of letting ourselves be drawn into some unseemly public pissing match. As a matter of fact, we earnestly hope that you will continue in your present position. The OSI does have its uses, and its glamorous leader is certainly an asset to Dexta."
"Then what do you want?" Gloria demanded. DuBray started to answer, but Gloria cut him off. "No, not _you._ I know what _you_ want. What about the rest of you?"
"We simply want your cooperation," said Grigsby. "In fact, we insist on it. Regardless of what the OSI Charter may say, all future OSI operations must receive prior approval from the appropriate Quadrant Administrator."
"You can't just come charging onto our turf like an avenging angel," said Chandra. "We will decide when a 'strategic intervention' is necessary and appropriate and when it is not. If you think you see a need for an intervention, come to us. If Norman assigns you to an intervention, come to us. If we agree that it is necessary, you can then proceed with our blessing and support. If we disagree, then there will be no intervention."
"What if Secretary Mingus orders me to proceed?"
"Leave Mingus to us," DuBray said.
"And what if I decide to ignore you and do what I think is best?" Gloria locked eyes with each of the Quadrant Administrators in turn.
"That would be foolish, dear," said Chandra, shaking her head. "You are young and idealistic, and Spirit knows we can use some of that around here. But don't imagine for a moment that your beauty and charm and courage somehow outweigh our institutional power. And while I can personally promise you fair and decent treatment, I'm afraid that I don't speak for everyone on that point."
DuBray smiled innocently. "Power exists to be used," he said. "You've already seen one aspect of the power at my disposal. If you continue to defy me, you'll see it again."
"Okay, this is where I get off." Manton Grigsby shot DuBray a dirty look, then walked past Gloria and out the door.
"And I, as well," said Chandra. She stepped forward and stopped in front of Gloria. "Although, I must say that there was a time when I would have stayed. You are remarkably attractive, although I think I could have given you a run for your money when I was in my prime." Chandra leaned forward and quickly whispered in Gloria's ear, "Fight dirty, dear. It's your only hope." She gave Gloria another smile, then left the room.
Gloria looked toward Algeciras and DuBray. Each man returned her gaze with a malicious half smile, like wolves contemplating their prey.
"I'll be going now, too," Gloria said. "I'll think about what you've said." She turned smartly and walked to the door, relieved to be making a quick exit. Her relief died stillborn. Waiting for her in the outer office stood a huge and grinning Erik Manko.
GLORIA STAYED IN HER OFFICE LATE THAT night, brooding. Her ribs still ached, but this time Manko had not beaten her badly, possibly in deference to DuBray's office furniture. He had merely punched her and thrown her a couple of times, enough to make his point. Then he had methodically stripped her, picked her up, and delivered her to DuBray and Algeciras in the inner office. Gloria was too frightened to try to fight Manko, and was discouraged by Chandra's words about the inevitability of it. Amid a wave of disgust and anger, she submitted to the Fours, as she had once submitted to the Fifteens. For her job. For Dexta.
"Fuck Dexta!" Gloria abruptly said aloud to the empty office. She had seen Dexta at its worst today, both personally and professionally. Four entrenched bureaucrats were waging a turf war against Norman Mingus, and Gloria and the OSI were simply pawns in their game. The fact that OSI had been doing good and necessary work did not enter into their consideration. The bureaucracy operated according to its own imperatives, and could not tolerate a maverick in its midst. It all made such perfect sense that Gloria even found herself agreeing in principle. The Quadrant Administrators had every right to feel affronted by the existence of OSI. Of course they would attempt to fight back.
More than that, Gloria realized that each of the four Quad Admins had reason to see her as a personal threat. Power within Dexta was a finite commodity; the more of it Gloria gained, the less would be shared by DuBray, Chandra, Algeciras, and Grigsby. If the OSI was a bureaucratic annoyance, its sexy and charismatic leader represented a potential rival for the ear of Norman Mingus and the opinion of Parliament and the public. So there was really no choice about it: The Quad Admins _had_ to oppose her.
But they didn't have to be such fucking pigs about it! Gloria wondered what it said about an organization when 50 percent of its upper management personnel were sadistic swine. Was that a requirement for the position, or was it something they learned on the job?
And yet, it was a job Gloria wanted. She wanted it so badly, it now seemed that she was willing to let smarmy jerks like DuBray and Algeciras have their way with her in order to preserve her position at Dexta. And why? So she could move onward and upward at Dexta until she became just like them?
"Fuck Dexta!" she said again, louder. "I don't have to do this, you know," she said to the silent walls, "I could be Empress."
Empress.
Coruler, more or less, of 3 trillion beings and a sphere of space two thousand light-years in diameter. An empire of 2653 worlds and counting. Empress.
Empress, as opposed to sexual plaything for depraved bureaucrats.
Algeciras had oozed over her like a bloated amoeba, dripping good intentions and romantic bile. He seemed to believe he was offering himself as a gift to her, and didn't understand why she would fail to accept it joyously.
DuBray had been less subtle. His choice of point of entry didn't surprise her, and she had simply leaned across his desk and accepted his pile-driving, battering-ram attentions without comment. From the sound of the muffled, urgent grunts he made as he went about his business behind her, she got the impression that DuBray really wasn't enjoying himself very much, either. This wasn't about sex. It was about domination.
"Fuck Dexta!" Gloria shouted it out.
Empress wouldn't be so bad, maybe. Gloria VanDeen-Hazar, Empress and Avatar of Joy.
Except that Empress was not the job she really wanted. Abruptly, she laughed out loud at her own audacity. It wasn't enough that she wanted to run the Empire—she insisted on running it on _her_ terms. Empress wasn't good enough; only Secretary of Dexta would do.
A sharp rapping at her office door was followed by the appearance of Arkady Volkonski. He stuck his head into the office and looked around. "I thought I heard a profanity directed against Dexta," he said.
"Stick around and you may hear it again," Gloria told him.
"I'd have to report you to someone," Volkonski informed her. "Disloyalty and seditious speech are to be discouraged."
"That's what I like about you, Arkady. I know I can always count on you to be a fascist."
"I wish you'd let me be one with Manko," Volkonski said.
"No!" Gloria barked. "I mean it, Arkady, I don't want you even thinking about Manko. I will not allow you to risk your career over this."
Volkonski frowned, but nodded. "Very well. But what _are_ you going to do about him?"
"I'm going to take the advice someone gave me today," Gloria told him. "Arkady, do you know how to get to Harlem?"
Volkonski shrugged. "Take the A Train?"
"YOU CAN'T COME IN WITH ME, ARKADY," Gloria told him as the skimmer hovered outside the entrance to Club Twelve Twenty-Nine. "I'll be perfectly safe in there, but you can keep a watch on the door if it makes you feel better."
"It would make me feel better if you didn't go in at all."
"It's just a club," Gloria assured him.
"Yes, and don't think I don't know what kind of club. Your personal life is your business, Gloria, but it bothers me—professionally and personally—to see you taking risks."
"Lately," said Gloria, "I've been taking a risk just getting out of bed in the morning. Relax, Arkady, I'll be fine. But I may be a while."
Gloria got out of the skimmer before Volkonski could raise any more objections. The bouncers welcomed her. Arnold was not behind the bar, but the woman on duty there directed her to a room to one side of the bar. She went in and found Arnold and a couple of other club employees lounging in some comfortable chairs, drinking beer, and watching a vid display of one of the null-rooms.
"Good evening, Arnold," Gloria said. "Do you suppose you and I could have a word in private?"
Arnold looked at the other workers and they dutifully exited. Gloria sat down in a chair next to Arnold. "You look better than ever tonight, Gloria," Arnold said. He reached for her and gave her upper thigh an affectionate squeeze. "What can I do for you?"
"I need a favor," she said. "I need to talk to a zamie."
Arnold leaned back in the chair and stared at her, clearly surprised. After a moment or two he nodded and said, "I can arrange that. Did you have anyone particular in mind?"
"I don't know from zamies," she said. "But I need to talk to someone as high up in the organization as you can manage."
"And would this conversation be of an official nature?"
"Call it semiofficial. It involves me personally, but there would be some Dexta interest. In any case, it would be off the record."
"Can I say what this is about?"
Gloria shook her head. "Only that it's a matter of some importance. I'd rather not say any more just now."
"All right," Arnold said. "I'll see what I can do. I assume I can reach you at Dexta?"
"Anytime."
"It may be a couple of days," he said. "I mean, I know some people, and they know some people...you know? Meantime, is there anything I can do for you tonight?"
"I was wondering if you have any Forty-eight yet?"
"Soon, they tell me. For now, all we have is Twenty-nine."
"In that case," Gloria said with a smile, "I think I'd like to try that wild cherry."
GLORIA TOOK THE NEXT MORNING OFF AND went to Rio. She wanted to talk to Charles. The Emperor, however, had ceremonial duties to attend to and would not be available for an hour or two. Gloria decided to kill the time with a walk in the Imperial Gardens.
She enjoyed the sights and smells of the botanical phantasmagoria, but was struck again by the symbolism of the place. Here, Terrans had deposited and nourished little bits and pieces of their Empire, green trophies of their triumphs. It seemed to Gloria that these biological oddities were really statements about the wealth and power of the Empire. We go where we want, and take what we want. Make careful note, O Faithful Subjects! We collected your plants, but we could just as easily have collected _you_!
Terran rule was not, by and large, cruel or oppressive. A few intransigent species had been wiped out during the course of mankind's expansion into the galaxy, but most had accepted the coming of the Earthers with varying degrees of acquiescence. Down through the centuries, the Empire had fought plenty of wars, large and small, and on some worlds there were still resistance movements and die-hard bands of guerrillas. But in the four decades since the defeat of the Ch'gnth, the Empire had mainly been at peace with itself, its subjects, and its neighbors. The era of peace was likely to continue indefinitely, since long-range probes had revealed no potential challenger to Terran hegemony within a thousand light-years of the Frontiers.
Contained within the Empire was a dazzling diversity of species, societies, and systems. Even leaving aside the alien civilizations, the human worlds of the Empire offered a cornucopia of cultures. Under the overarching rule of the Emperor, Parliament, and Dexta, individual worlds and small confederations had been free to work out their own systems of government and social organization. As long as they respected basic sentients' rights (at least in the abstract) and didn't overtly challenge Imperial rule or Dexta's ursine embrace, such worlds were welcome to develop as they would. Over 60 percent of human-inhabited planets operated under some form of democratic rule, but the others featured everything from feudal kingdoms and ancestral satrapies to communist collectives and fascist dictatorships. Seventy percent of the Empire's residents called themselves Spiritists, yet there were also enclaves where strict Muslims, Hindus, and Christians held sway. The Jews had worlds of their own. So did the French.
All things considered, it was a rather relaxed empire—the only kind possible, really, given the distances and numbers involved. It was a strategy that had worked well—the Terran Empire had already lasted longer than those of Rome, Britain, or America. The Empire didn't try too hard to impose its will as long as its subjects didn't try too hard to resist it. People like the anarchists of PAIN objected to the very principle of empire, but there was no simmering cauldron of discontent for them to tap into. The O'Neill Dictum—"all politics is local"—applied even on a galactic scale. The issues that bothered people in Quadrant 1 were unlikely to concern those a thousand light-years away in Quadrant 3. Ruling the Empire, it seemed to Gloria, was mainly a matter of preserving the natural equipoise and inertia that governed any such immense entity.
Ruling the Empire! She could do it, or half do it, with a single word to Charles. She _could_...
And yet, ruling the Empire was not quite the same thing as _running_ it. Charles ruled the Empire, but Norman Mingus had more tangible power at his fingertips than did Charles, or any Emperor since Hazar the Great.
But what had Grigsby said? Mingus doesn't run Dexta, _we_ do? As Charles ruled the Empire, Mingus ruled Dexta, but the Quad Admins actually ran it. Operational, day-to-day power within Dexta was concentrated in the grimy hands of DuBray, Chandra, Algeciras, and Grigsby. And that was not likely to change. What would that mean for the OSI?
Running the OSI was fun. It gave her the opportunity to dash all over the Empire and solve problems that didn't really matter very much in the grand scheme of things but deeply affected the lives of those involved. The dispute on Cartago had been barely a notch above trivial, and yet the people of that barren world were probably going to lead at least slightly better lives because of what Gloria had done. That was something to be proud of, and she was. And aside from the fact that now and then people tried to kill her, Gloria enjoyed her OSI missions.
There was an intense, almost sexual thrill about it all, like an extended ride on Orgastria-29. It was like screwing on a roller coaster, in full view of 3 trillion people.
Gloria smiled at the thought and found herself singing one of the old twentieth-century ditties she enjoyed: "And I'll have fun, fun, fun, till my daddy takes my T-Bird away!" She wasn't too clear on what a T-Bird was, but she liked the concept.
And she could go on having that kind of fun—until the Quad Admins took her T-Bird away. Gloria kicked at a pebble on the walkway and sent it flying. OSI was now all but officially under siege by the Quad Admins. She could hang on by her fingernails and try to keep the OSI independent and alive—but it was a battle that promised to be grim and costly.
Or, she could become Empress.
She held out her hands, palms up, and mentally weighed her options. In her left hand she put the advantages of being Empress. Immense power. A huge responsibility along with it, true, but still...
There were the perks. Having been wealthy all her life, Gloria seldom thought about money, but she did appreciate the luxuries it could buy. As Empress, she could have every luxury imaginable, and then some. Her left hand sagged a bit lower from the weight of the thought.
But would it really be... _fun, fun, fun_? Surrounded by Imperial Security, hounded incessantly by the media, some sort of living goddess to the adoring but demanding masses? All of that had a certain appeal to her, she had to admit, but it might become tiresome after the first twenty or thirty years.
In her right hand, she put Dexta. Dexta, _dexter;_ Empress, _sinister._ Had she meant something by that?
Dexta...all the _fun, fun, fun_ a girl could ask for. Like being terrorized by the Pack Dogs and her superiors when she was a Fifteen. Like having people try to kill her. Like being butt-fucked by Cornell DuBray. Like having to face the ugly little realities of life in the glorious Empire—attempted genocide on Mynjhino, and the smell of the fires in the park, where the Jhino troops were cremating the bodies of thousands of Myn. Or the camp at Pizen Flats on Sylvania, strewn with the remains of people who had been her friends.
But in the end, she had been able to redress the balance on those two worlds, though not without cost. She had made a difference. The Empire was a marginally better place than it would have been without her. That counted for something.
On the other hand, consider how much more she could do as Empress. And as Empress, she would have no Quad Admins standing in her way. She would not have to deal with Cornell DuBray. Only Charles...
She stared at her hands for a few moments, then threw them up in the air in defeat. She was damned if she knew _which_ hand Charles should go in.
She had been young and stupid when she married him. Was she still young enough and stupid enough to marry him again? Spirit, hadn't she learned anything? Of course, it was different now. Back then, Charles had not been Emperor and was unlikely ever to be; she was simply his wife. And being his wife had not been all bad, in honesty. Charles, who had the male version of the enhanced genes Gloria boasted, was probably the best bedmate she'd ever had. And in spite of being an essentially selfish bastard, there were times when he had shown some dim signs of intelligence, compassion, and generosity.
Could she rule the Empire at his side? Have a son with him?
Gloria found herself standing in front of the gnarled glashpadoza tree, the one that absorbed the sins of its owner. An appropriate item for an emperor to have. And what, she wondered, would Dexta's glashpadoza tree be like? How big, how ugly would _that_ one be?
At least she didn't have to decide between them at once. No, she could brood about it for weeks and let it eat up her life, instead.
_Fun, fun, fun,_ she thought.
GLORIA'S ROUTE TO THE RESIDENCE TOOK HER past the swimming pool, where she found an unpleasant surprise. Laurence, Lord Brockinbrough—Charles's detestable cousin and heir—was camped out poolside, shielded from the fierce Rio sun by a purple robe and a broad-brimmed straw hat. He raised a hand to her and waved it idly. "Gloria," he called. "Grand to see you again, milady. Or should I say, my Empress?"
Gritting her teeth, Gloria managed a smile and a cool, "Hello, Larry."
It had been seven or eight years since she had seen him. He had put on weight. There had always been something snakelike about him, Gloria thought, but now he more closely resembled one of those plump lizards that lie around on rocks and wait for a meal to walk by. He was in his fifties and could have looked thirty, but the additional weight had pushed him into some nebulous zone of indeterminate middle age. His cheeks were full and round, his lips fleshy, his eyes still dark and impenetrable.
"I intend to be best man again," Larry told her, smiling impishly.
"Don't get your hopes up, Larry," Gloria replied. "Nothing has been decided."
"Playing the reluctant virgin, are we? I have to say, the role ill becomes you. I know you better than that, after all."
He knew her entirely too well. She and Charles had spent much of their marriage roaming the Empire in his luxurious yacht, and on some of those jaunts, they had been joined by Cousin Larry. Joined, even in bed. Charles enjoyed experimentation.
In those days, Charles had been nothing but a callow and superfluous member of a fecund dynastic clan, far from power or the prospect of it, and Larry an older and even more superfluous embarrassment to the Hazars. Somehow, they had latched on to each other and become what they described, in drunken giggles, as coconspirators in a Plot Against the Empire. They both affected a pose of being cynical young rebels, contemptuous of the dynasty that imprisoned them, although Larry was already getting a little old to be credible in the part. Charles had seemed to be studying his cousin, picking up pointers for what promised to be a life of idle merriment and utter uselessness.
A loud splash from the pool drew Gloria's attention. She saw a naked young man cavorting there with two equally naked young women.
"You remember my son, Gareth, of course," said Larry.
"That's Gareth?" Gloria asked in genuine surprise. She remembered him as a ten-year-old brat. He was the product of Larry's union with his first wife. The wife he had beaten to death in a drunken rage.
Such things happened, of course, even in the best of families. It was the media's _scandal du jour_ for a time, but then Larry was packed off to a Rehabilitation Clinic for a year, and no more was said about it. And nothing at all was said about the second woman he killed, an anonymous nobody on some distant outworld. A little more rehabilitation, a little less booze, and Larry was once again fit to be seen in public.
"I wouldn't have recognized him," Gloria said, looking toward young Gareth in the pool.
"He had a rapid and successful puberty," Larry said. "He's eighteen now. Smart as a whip, too."
"Just a chip off the old block, huh?"
Larry smiled. "I'd like to think he's better than that."
"He'd almost have to be, wouldn't he?"
Lord Brockinbrough gave a minimal shrug. He must have been well aware that Gloria despised him; most people who knew him did. He simply refused to let that bother him.
"Gareth!" Larry called. "Come on over here and say hello to Gloria."
The young man dutifully swam over to the side of the pool, hauled himself out of it, and stood before Gloria, naked and dripping water from various appendages. He was tall, slim, and well formed, with the same serpentine eyes as his father. "Hi, Gloria," he said happily. "Been a long time." His eyes made a slow and deliberate inventory of everything revealed by Gloria's flimsy wrap dress, which was nearly all of her.
"Hello, Gareth. You've grown up, I see."
He glanced down at himself, then looked into Gloria's eyes and grinned. "Yes, I have," he said. "You might even like me now. Are you going to marry Charles again?"
"That remains to be determined," Gloria said formally, wondering if everyone in the Household was aware of Charles's proposition.
"Well," Gareth said, "if you don't, _I'm_ available." He winked at her.
"I'll make a note of that," Gloria told him.
"No, really," Gareth persisted. "If marriage is out, maybe we could just fuck sometime. I mean, I know you did it with my dad, and I'm better than him. Isn't that right, girls?" he called to the young women in the pool. "I'm better than my dad, aren't I?" The girls grinned and nodded enthusiastically.
Gloria glanced at Lord Brockinbrough. "Is there a gene for narcissism?" she wondered aloud.
"Back in the pool, son. You're embarrassing Gloria, who is a woman of refined sensibilities and lofty pretensions. Anyway, you're getting a hard-on, staring at her like that."
Gareth laughed. "Seeya later, Gloria!" He dived into the pool and porpoised his way back to the girls.
"All the grace and charm we've come to expect from the Brockinbroughs," Gloria observed.
"He still has a few rough edges," Larry admitted, "but I'm proud of him, I truly am. He may yet redeem the family name that I so thoughtlessly besmirched. I have great hopes for him."
"Hope is a wonderful thing," Gloria agreed.
"Charles has high hopes, too," Larry said. "Don't disappoint him, Gloria. Marry him. Give him a son. As things stand now, _I'm_ his heir. I don't like that any more than the rest of the House of Hazar does. We would all be thrilled and gratified if you were to become Empress and relieve me of the burden and embarrassment of being next in the line of succession."
"I honestly haven't made up my mind, Larry," Gloria said.
"He needs you, Gloria. More than he realizes. He was devastated when you left him."
Gloria responded to that with a loud horselaugh.
"Well, he was," Larry insisted. "He may not have shown it, but he was deeply hurt."
"I seem to recall," Gloria said, "that after I left him, the two of you went off on an inspection tour of all the bordellos in Quadrant 3."
"It was merely a way of coping with his grief. Charles is deeper than you ever gave him credit for. He's an intelligent and caring man, but given his upbringing, he's always felt that he had to hide his inner self. Growing up as a Hazar is more of a burden than you can fully comprehend, Gloria. I never managed to overcome it, but Charles has, and will only improve with age—with you at his side."
Gloria didn't say anything in response. But she wondered if he could be right.
CHARLES WAS FREE, AND GLORIA WAS SENT into his private chambers. There, she found him changing out of his Imperial garb and into a tee shirt and a pair of old, faded jeans.
"Where does an Emperor get worn-out jeans?" she asked him.
"He simply keeps the same pair for ten years and threatens the catacombs for anyone who dares to mend, repair, or refurbish them. I had these jeans when we were together, Glory. I can still fit into them, too, thank you very much."
"One way, at least, that you don't take after Cousin Larry."
"Saw him, did you?"
"That misfortune was mine, yes."
"He's not as bad as you think he is."
"He said precisely the same thing about you."
"Really? Well, there, you see, that proves it's true for both of us. Now, what can I do for you? The Emperor is entirely at your disposal."
"Will I talk about myself in the third person, too, if I become Empress?"
"Probably," Charles said after a moment's thought. "There are times when I think of the Emperor as some other guy, some distinct entity, different from _me._ "
"So, you take off your robes and garters and whatnot, get into your jeans, and suddenly you're no longer the Emperor? Is it as easy as that?"
Charles shook his head. " _Nothing_ is easy for an emperor. But you get used to it. I did, and so will you."
"Maybe." Gloria sat down on the edge of the bed next to Charles as he put on a pair of sneakers. "What will it be like?" she asked him. "For us, I mean?"
"As husband and wife? Or as Emperor and Empress?"
"Both. You weren't exactly an ideal husband the first time around."
"I suppose we can take that as a given. Perhaps we can also assume that I learned something from the experience, and that I won't be quite as much of a cad in the future as I have been in the past. I expect that we would both retain a certain degree of freedom, however, if that's what's on your mind. Far be it from me to cramp the style of an Avatar of Joy."
Gloria nodded. She had to admit that marital fidelity was not a concept that worked well for either one of them. She had tried to be faithful to Charles after they were first married, but when it became obvious that he had no intention of cleaving only unto her, she had sown a few wild oats of her own. Well, more than a few, actually. And Cousin Larry was hardly the only person to have shared a bed with both of them.
"Mind you," Charles added, "I don't think it would be appropriate for the Empress to start humping beggars in the streets. But short of that, I think you'd probably be able to have as active a social life as you desire. No more trips to the Club Twelve Twenty-Nine, though. But I've set up a null-room here in the Residence, and as for drugs—"
"Dammit, Charles, you're having me followed!"
"Let's just say that I'm keeping tabs on you. For instance, I know that Cornell DuBray is giving you a hard time. And that Manko hominid...say the word, Glory, and I'll make him disappear."
"I can handle Manko myself," she insisted. "And DuBray, too."
"But why should you have to?" Charles asked in honest bafflement. "That's the part I just don't understand. Never have. What the hell are you trying to prove? You're rich, beautiful, and privileged—you always have been and you always will be. Why the pretensions of being just another working girl, carrying her lunch in a brown paper bag?"
Gloria snickered. "I'm hardly _that._ "
"No, but you seem to want people to think that you are—when you aren't busy being an Avatar of Joy. It seems to me that you're trying to have things both ways, Gloria. You want to be an honest, dedicated bureaucrat, a true servant of the people, but at the same time you want to go on enjoying all the advantages of your wealth and position."
"I don't see why I can't do both. Being the glamorous Gloria VanDeen—you know, sometimes I think of _her_ in the third person—gives me a certain power, both inside and outside Dexta. I use that power to do my job. You may not like some of the things I've done, but you can't deny that I've been pretty effective."
"But think how much more effective you could be as Empress." Charles seized her left hand and squeezed it. "Think of all the good you could do for humanity!"
Gloria abruptly pulled her hand away from him. "Oh, please," she groaned.
Charles grimaced. "Okay," he said, "I suppose that one doesn't work very well, coming from me. We both know that I care more about the well-being of my dogs than I do about the general run of humanity. As long as they aren't rioting in the streets for bread, I'm not too concerned about their happiness. Spirit, Gloria, there are _three trillion_ of them. My job is to be concerned about the welfare of the anthill, not the ants. But if you _do_ care about them—and I believe that you sincerely do—then here's your chance to do something for them."
"Like what?"
"Entertain them, for one. They need that, you know, almost as much as they need the bread. Bread and circuses, like in Rome. The plebs haven't changed in three thousand years, Gloria. They still want sex and spectacle, and who better to provide that than you?"
"I give them that already," Gloria pointed out. "What more would I be able to do if I were Empress? You talked about real power, Charles. What, exactly, are you offering?"
Charles got to his feet and wandered out onto a balcony. Gloria followed him and took in the green opulence of Rio and the looming bulk of Sugar Loaf. The central spire of the Spiritist Mother Church impaled the nearby sky. Far below them, she noticed the swimming pool, where Cousin Larry and son Gareth seemed to be busy having sex with the two young women she had seen.
"Sex and spectacle," she mused. "But what about power?"
"Within reason," Charles said at last, "I can see giving you authority over virtually anything you want, with the exception of the military and the economy."
"Those are pretty big exceptions, Chuckles."
"And necessary ones. The military can have only one commander-in-chief, and that must be the Emperor. Surely, you can see that. As for the economy, in a general sense, it runs itself. In a more specific sense, policies such as taxation, money supply, and resource development all involve a complex set of relationships among the Household, Parliament, and the Big Twelve. It wouldn't do to have you meddling in that miasma. You would find it stupefyingly boring, in any case."
"So what's left?"
"Social welfare, in all its many guises. Food for the starving, clothes for the needy, books for the ignorant. You could play an important role in seeing to it that the Empire is not populated exclusively by hungry, naked morons."
"If you control the economy," Gloria pointed out, "that would leave me distributing crumbs."
"What you get would depend on how well you play the game. It's nice to pretend that as Emperor I can just snap my fingers and get what I want, but that's not the way it works. I still have to fight for the programs and policies I favor, and I don't always get my way. Neither would you. But as Empress, you would carry considerable weight with the public and, therefore, with Parliament. What you ultimately achieved would be determined by your own abilities."
"What if some of my social welfare issues overlapped with your domain?"
"Such as?"
"The Big Twelve dominate some worlds in ways that go far beyond mere business. In some places, they are virtual feudal overlords."
"Aren't they just?" Charles said with a snort. "And you think you can do something about it?"
"I'd like to try," Gloria said.
Charles thought about it for a moment. "A ticklish area," he said at last. "But it strikes me that it might be useful if we could double-team them. 'Sorry, GalaxCo, I'd love to help you out, but it would upset the Empress.' That sort of thing. Together, we'd have some leverage with them." He turned to her and grinned. "It might even be fun."
"There are some other things," Gloria said. She pointed toward the sky. "There's still slavery out there."
Charles nodded. "I know," he said.
"Back on Mynjhino, Randall Sweet talked about selling me to the royal family on Shandrach. What if I wanted to do something about those bastards?"
"You'd have my cautious support. I'm not Lincoln, and I have no intention of having an Antietam or a Gettysburg on my watch. But I don't much care for the slavers, and I wouldn't mind if the history books called me Charles the Emancipator. Sounds pretty good, don't you think?"
"So what does 'cautious support' mean, exactly?"
Charles sighed. "I guess it means that I'd be with you in principle, but that there are limits to how far I'd be willing to go in practice. I'm not going to let you drag me—and the Empire—into a crusade or a civil war over the issue. One thing you'll understand as Empress, Glory, is that you have to take the long view. Some problems may take decades to solve. Others may take centuries. Still others may have no solution at all. You have to be able to tell the difference and act on that understanding."
"I appreciate what you're saying, Charles, and I don't expect miracles. But you've been pretty vague about all of this."
"You want specifics? Very well, I'll consult with the appropriate people on my staff and draw up some sort of document laying out the details of our joint rule. You'll object to it, of course, and propose changes. We'll change the changes, and you'll change the changes to your changes. Eventually, we'll come to an agreement. Then we'll have something to wave in each other's face after we've violated the agreement."
"All right. Put your people to work on it."
Charles turned to face her and put his hands on her shoulders. "We're really going to do this, then?" he asked.
"We might," Gloria said airily, as if they were discussing having lunch. "I mentioned this to Mingus. He said we might do a great job together as Emperor and Empress."
Charles's eyebrows rose. "Norman said _that_? Well, then...what better endorsement could you want?"
"He also said that we might wind up leading the Empire into a civil war."
Charles laughed. "The Charlesists versus the Glorianos? What fun! If I end up chopping off your head, I promise you I'll have it preserved and tastefully displayed in the Imperial Museum."
"And I," Gloria assured him, "will do the same with whatever part of you _I_ chop off."
GLORIA STOOD ON THE SIDEWALK AT THE MAIN entrance to the immense Dexta complex, which took up twelve city blocks' worth of ground space in midtown Manhattan. The day was raw and drizzly, and she wore a buff-colored trench coat over her navy miniskirt and jacket. People walking by stopped and stared at her briefly, then continued on their way; Manhattanites had always been accustomed to celebrities in their midst.
A long black limo pulled over to the curb and hovered there. A door popped open and Gloria stepped in. She sat down next to a man in an expensive overcoat as the door closed behind her and the skimmer pulled away from the Dexta complex. Without bothering to look, Gloria knew that behind the limo, another skimmer filled with Bugs was following at a discreet interval; she had told Volkonski to keep his distance today.
"Ms. VanDeen?" the man said. "I'm Ed Smith. Pleased to make your acquaintance." Gloria nodded politely and shook his hand. He was a prosperous-looking man of about sixty, she guessed, a little overweight but with a healthy, ruddy look to his face.
"Smith?" Gloria couldn't help asking.
The man shrugged. "Okay, so I'm not Ed Smith. But you have to call me something, so it might just as well be Ed Smith, right?"
"Fair enough," Gloria said. "Thank you for agreeing to see me."
"It wasn't exactly a hardship," Smith said, grinning. "When I got the request, it took me about a nanosecond to clear my calendar. I've seen you in person before—here and there, around town—but I never really expected to have the pleasure of taking you to lunch. By the way, we'll be going to a little French restaurant we own over on the East Side. There's a private room, so we won't be seen or disturbed."
"French?" Gloria asked. "I have to confess, I was expecting Italian."
"Or maybe Russian? Or Japanese? Stereotypes die hard, I suppose. The fact is, I don't have a drop of Sicilian blood in me. The organization of which I am a part is—how shall I put this?—ecumenical."
"And if I could ask, just how did you become a part of it?"
"Believe it or not, I was recruited out of college," Smith said with a slightly embarrassed smile. "I ran the football and basketball betting pools at my fraternity. I was good at it, and pretty soon I was handling the sports book for every frat on campus. And when I graduated, instead of going with GalaxCo or Imperium, I accepted an offer from the zamitat. And I've never regretted it."
They arrived at the restaurant; Gloria mentioned that she had eaten there several times. "You do business with us every day, Gloria, whether you realize it or not," Smith told her as they got settled at a table in a private room. Since he seemed to know the place so well, Gloria let Smith order wine and their meal.
"This restaurant and places like the Club Twelve Twenty-Nine are only the tip of the iceberg," Smith explained. "We like to think of ourselves as the thirteenth of the Big Twelve. We have long-term relationships with each of them, as well as with Dexta, Parliament, and the Imperial Household itself. The zamitat provides important services that people want and need and are not otherwise available. We are a necessary social lubricant."
"I see," said Gloria. She knew little about the zamitat and was interested to learn how they viewed themselves.
"Governments," said Smith, "have sometimes felt it necessary—or politically expedient—to outlaw things like prostitution, gambling, drugs, alcohol, tobacco...whatever. But the demand for them never goes away. Organizations like the zamitat and its predecessors bridge the gap between what is politically desirable and what is socially necessary."
"Last year," Gloria said, "I found myself running a bordello on Sylvania. All perfectly legal, of course, but if it had been illegal, I don't think that would have made the slightest difference to our operation."
"Of course not. People want what they want, and other people have always found a way to provide it for them. As I said, we offer a necessary social service—no less than Dexta."
"Speaking of Dexta..."
Smith shook his head. "If you are not already aware of the nature of the links between the zamitat and Dexta, I am certainly not going to be the one to tell you. Given your position as head of the Office of Strategic Intervention, you might conceivably feel compelled to take action on such knowledge if you had it."
"Just a thought." Gloria shrugged. "Anyway, I didn't want to see you about that. It's something else entirely."
Smith smiled at her pleasantly. "How may we be of service?" he asked.
Gloria locked eyes with him. "Someone is trying to kill me," she said.
"That business recently on Cartago? Yes, I heard about that."
"There are reasons to think that PAIN was responsible. The attack on me seems to tie in with two terrorist episodes in Quadrant 4. Our security people are concentrating on that possibility. But I wonder if there might be other possibilities."
"Such as?"
"I've annoyed and frustrated a lot of people in my work at OSI. Including some of the Big Twelve."
Smith nodded. "And you think one of them might have taken out a contract on you?"
"That's one of the 'necessary services' the zamitat provides, isn't it?"
Smith stroked his chin for a moment, then took a sip of wine. "Interesting notion," he said. "But I can tell you that I've heard nothing about any such contract."
"And would you know, if one existed?" Gloria asked him.
"Not necessarily," Smith conceded after a moment's thought.
"Could I ask, just what is your position in the organization? I mean, how high up are you?"
Smith smiled a little. "I suppose," he said, "you could think of me as a vice president, of sorts. But that's really somewhat misleading. You see, the Big Twelve corporates are all vertically integrated and organized. There are distinct boxes on the organizational chart, leading upward to a well-defined hierarchy of leadership positions. With the zamitat, the organization and integration are more horizontal. We are, in effect, a cartel—a voluntary association of semiautonomous groups, which we refer to as divisions. Historically, they were known as 'families,' although blood ties and ethnic identity now have little or nothing to do with it in most cases. The divisions are distinct entities, but they cooperate with one another rather than competing. When disputes arise, they are resolved by a sort of upper-level advisory board that has real but strictly limited powers. We have nothing comparable to a corporation's board of directors, and no chief executives."
"So what you're saying," Gloria attempted to summarize, "is that if some other division of your organization had agreed to a contract on me, you might not be aware of it."
Smith nodded. "Still," he said, "I would think that such a contract would be unlikely. I don't think any of our divisions would want to be involved in an attack on the most famous and popular woman in the Empire. There would have to be some motivation beyond mere money. Have you done anything that would seriously annoy any of our people?"
"Not that I know of," Gloria said. "Can you tell me, would the zamitat accept a contract from PAIN?"
"We've had dealings with them in the past," he said with a slight shrug, "but I'm not aware of anything like what you're suggesting. On the face of it, that seems unlikely to me. That assassin on Cartago was killed, wasn't he?"
"Instantly," said Gloria.
"Well, there you are. If it had been one of our people, he would not have let himself be killed, and you and I would not be enjoying each other's company today. What the zamitat offers, Gloria, is professionalism, in everything we do. The attack on you was clearly the work of an amateur."
Their meals arrived, and they dined in thoughtful silence for a while. At length, Gloria decided to explore another angle.
"There's something else," she said. "The attempt on my life, and at least one of the two terrorist attacks, involved the use of old Mark IV plasma rifles. And each of those rifles was part of a shipment to Savoy in 3163, just before the start of the war with the Ch'gnth."
Smith's eyebrows rose slightly. "You don't say?"
"Would the zamitat know anything about that?"
Smith pursed his lips, then said, "Before my time, I'm afraid."
"Mine too," said Gloria. "But those weapons should have been destroyed in the attack on Savoy in 3163. Yet, suddenly, here they are. It occurred to me that perhaps some of the weapons earmarked for the Savoy shipment were...shall we say, diverted. That's the kind of thing the zamitat does, isn't it?"
Smith drummed his fingers on the tablecloth for a moment, then nodded. "Such things have been known to happen," he admitted. "If I remember my history correctly, the Savoy colony was pretty much wiped out, wasn't it?"
"That's right."
"Then those weapons either never made it to Savoy or were somehow salvaged from the wreckage. I don't have any personal knowledge of this, but either way, zamitat involvement would not be improbable. I know the organization turned a substantial profit in the immediate postwar years simply by salvaging and reselling leftover ordnance. And, of course, we've always had substantial involvement in shipping and...um...diversions. But, again, I simply don't know."
"I understand," Gloria said. "But could you find out?"
Smith moved his head in an ambiguous side-to-side, up-and-down motion. "I could ask around. Some of our senior people might know. But remember the way we are organized, Gloria. Earth's involvement in some enterprise out in Quadrant 4, fifty or sixty years ago, may well have been pretty minimal. And even if there was some zamitat involvement in the Savoy business, it doesn't mean that there is any involvement in the current matter."
"I understand," Gloria said. "But at this point, _any_ additional information would be welcome."
"Your people have no leads? That surprises me. The Bugs are usually pretty efficient."
"It's an odd case," Gloria pointed out. "They don't get many that have roots going back half a century."
"Neither do we," Smith said. "But I'll see what I can find out for you. I assume you can be reached at Dexta?"
"For now," Gloria said. "But next week, I take off for the Quadrant Meeting on New Cambridge."
"No problem," Smith said with a grin. "We'll be there, too."
Gloria ate in silence for a while. Zamitat help with the Savoy weapons might turn out to be useful, although it was something of a long shot. But that wasn't the real reason she had wanted to see Smith.
To this point, the contact had been clean and professional—chaste, in a way. But the next step, if she took it, might put her on a road from which there could be no turning back. She would be establishing a relationship. And it wouldn't be free.
Smith seemed to sense her discomfort. "Is there something else we can do for you, Gloria?" he asked.
_A necessary service,_ she thought.
"Yes," she said at last. "I need a favor."
GLORIA WAS CURLED UP ON A COUCH IN HER penthouse, munching carrot sticks and reading a history of the war with the Ch'gnth when the text on her pad was overridden by a vid image from the downstairs desk. "There's an Eli Opatnu here for you, Ms. VanDeen," the guard at the desk informed her.
"Send him up," Gloria said. The news both surprised and pleased her. She put the carrot sticks away in the kitchen, then went to the front door. She was wearing only a very thin, blue silk robe, knotted at the waist. After a moment's hesitation, she loosened the knot and pulled the fabric farther apart.
Opatnu gave her a toothy grin at the door and held up a bottle of wine. "A peace offering," he said.
"Everyone keeps giving me wine lately," Gloria said, ushering him in. "Tell you what. Hold off on the wine for a bit, and I'll fix us something better. Go on into the living room and make yourself comfortable."
"Whatever you say." Gloria watched him as he walked, taking note of his long, lean form and the graceful, unhurried rhythm of his movements. Plus, he had a great ass.
She popped a porcelain kettle into the 'wave and nuked it for a few seconds, then poured the contents into two big mugs. A moment later she arrived in the living room, handed one mug to Opatnu, then sat next to him on the couch. She raised the steaming mug, clinked it against his, and took a sip. He did the same.
"What's this? Some kind of tea?"
"Sort of," Gloria said. "It's from Mynjhino. I think you'll like it. Maybe you can open the wine a little later." Gloria drew her legs up and scrunched around on the couch. The movement was enough to cause the loose robe to fall away to the sides, uncovering her breasts. Opatnu took note, and Gloria noted his note-taking. Her skin was already tingling, and she didn't think it was just the jigli tea. She wondered if Eli might have some genetic enhancements of his own; some people had the ability to pump out pheromones on demand. Whatever the cause, she was suddenly feeling exceptionally randy.
"I thought I should apologize for what happened in your office," Opatnu said. "I just marched in there and expected you to tell me everything I wanted to know. It didn't occur to me that you had imperatives of your own. I hope my thoughtlessness hasn't irreparably damaged our relationship."
Gloria smiled slyly. "Oh," she said, "I have a feeling it will recover."
Opatnu grinned and stared at her bare breasts for a moment. "Oddly enough," he said, "I have that same feeling." He looked down at his steaming mug. "Say, what exactly is this, anyway?"
"You like it? It's called jigli. It is reputed to be the most powerful natural aphrodisiac in the Empire."
"Really? Funny, I've never heard of it."
"Almost no one has. You see, the Myn don't grow it for commercial use or export—that would violate their religious scruples. But I just happened to acquire some while I was on their planet."
"Interesting," he said, and took another sip. "Very interesting...where was I?"
"You were making a totally unnecessary apology," Gloria reminded him. "You had every right to be curious about an OSI intervention in your sector. I was going to brief you soon anyway, so I shouldn't have been so persnickety about those mysterious sources of yours. I'd still like to find out who they are, of course, but I won't press you on it. Not tonight, anyway."
"Thank you, Gloria. I appreciate that."
"As for our investigation, it concerns what is known, historically, as a double-flagging operation."
Opatnu nodded. "Freighters with multiple identities, that sort of thing?"
"Exactly," Gloria affirmed. "It works a couple of different ways. In one, you get yourself two identical freighters and give them the same name and duplicate sets of registration papers. The two freighters work in different places simultaneously, and pay whatever local tariffs apply, but only one of them pays the Imperial taxes. In another version, you get two or three sets of papers for a single freighter, then change its identity as appropriate to evade local taxation."
"Uh-huh," Opatnu nodded. "And you think this is happening in my sector?"
"We've received some preliminary indications of it," Gloria said. "I have a Financial team on Staghorn at the moment, checking to see if there's any substance to the charges. Of course, you must realize that in order for a scheme like this to work—"
"There would have to be Dexta involvement. You think I'm running a corrupt sector?"
"I don't think any such thing," Gloria insisted. "But corruption does happen. We don't care if people are cheating on their taxes, but if they are doing it with help from someone in Dexta, that makes it an OSI matter."
"I suppose so," Opatnu said with an air of glum resignation. "Look, Gloria, I like to think that I run a pretty clean shop, but you know as well as I do that there's corruption of one sort or another in all twenty-four sectors. But there are some things you need to understand about _my_ sector."
"Like what?"
"Sector 19," Opatnu began, "has the second-lowest population density of any sector. About forty percent of our territory was Ch'gnth space, before the war. Expansion and settlement were blocked for centuries. In consequence, we only have eighty-six colonies in the sector—far below average. But in the decades since the war, all that territory has opened up for us."
"And you now have the highest growth rate of any sector," Gloria put in.
"Right. Something to be proud of, but it also presents problems that are unique to Sector 19. Because of our low population density, the Big Twelve have never been very interested in doing business in the sector. Too much empty space, too low a rate of return. Postwar expansion has attracted them, of course, but mainly to the Frontier region. The older, occupied sections of the sector are still underserved by the big corporates. And so, a lot of the sector's transport and commerce is in the hands of smaller local and regional concerns."
"Like Wendover Freight and Storage?" Gloria ventured.
"Wendover?" Opatnu gave her a sharp gaze. "You think Wendover is doing the double-flagging?"
"Possibly," Gloria said.
"Well," Opatnu said, "I suppose they'd be the logical candidate. They've been the major shipper in the sector for a century or more. And it may be that they decided they need to cut some corners in order to stay in business these days, given the unique circumstances in the sector. We've got a pretty good boom going on in the outer regions, which are dominated by the Big Twelve, and at the same time, we've got long-term stagnation in the rest of the sector, where Wendover does its business. But Gloria, if Wendover is guilty of something like this, and you take them out, that could turn stagnation into outright recession, or worse. I'm not defending anyone's criminal activity, understand, but I hope you'll keep the big picture in mind as you pursue this."
"Well," Gloria said, "I suppose I'll just have to rely on you to keep reminding me of the big picture." She leaned close to him and took his face in her hands. They locked eyes for a second, then plunged into a jigli-fired kiss that quickly evolved into eager groping.
Opatnu untied Gloria's robe and stared for a moment at her smooth, cocoa-colored flesh, the dark prominences of her straining nipples, and the honey-blond curls of her pubic mound. Meanwhile, she unbuttoned his shirt and ran her hands over the taut, almost hairless expanse of his chest. Then her hands descended to his belt, unfastened it, and opened up his trousers.
"Speaking of the big picture...!" she exclaimed.
"Like I said," Opatnu told her, "my sector is noted for its rapid expansion."
Opatnu thrust himself into her, and Gloria cried out in delight. He was, she decided on the spot, as skilled and impressive a lover as Charles. She spasmed in electric rapture, and only in the very back of her mind did it occur to her that the moment would have been even better if she had a little Twenty-nine.
"I JUST WANTED TO LET YOU KNOW THAT I'LL BE leaving tomorrow for New Cambridge and the Quadrant Meeting. I suppose I'll see you there."
Gloria stood in front of Cornell DuBray in his plush office, wearing a nearly transparent shirtdress, minimally buttoned. She wanted to give him a good look at her body—the body he would never again touch.
"I guess Erik Manko won't be making the trip with you, will he?" Gloria added.
"What do you know about that?" DuBray snapped at her.
Gloria shrugged innocently. "Only that he was attacked and badly beaten. Right here on the streets of Manhattan. Shocking, isn't it, that such things still happen? But I'm sure he'll recover, in time."
"A month," DuBray snarled.
"Well, I hope he'll be more careful from now on. It would be terrible if something like that happened to him again."
"You've gone over the line, VanDeen," DuBray growled.
"Which line would that be?"
"You went outside of Dexta to settle a personal matter. You couldn't handle Manko yourself, so you hired some muscle to do the job for you. I've half a mind to have Internal Security look into this."
"Look into what? Manko's misfortune happened out on the city streets. It's a matter for the New York City Police Department, not IntSec. They have no jurisdiction."
"They do if I tell them they do. Don't get too clever with me, VanDeen. You'd regret it."
"More threats and bullying, DuBray? Where's all that upper-level finesse you're so proud of?"
"I can break you, VanDeen. Anytime I want."
"Would you like to try it here and now?"
DuBray offered her a thin, cold smile. "We are of unequal levels, as you are well aware. Don't think you can provoke me into physicality. I have other means at my disposal."
"You might give a little thought to some of the means at _my_ disposal," Gloria suggested. "I'm not some anonymous Ten cowering before the awful power of a Four. The public adores me, Norman Mingus has faith in me, and I have powerful friends and relations—and _ex_ -relations."
That gave DuBray pause, as Gloria had intended it should. This was the first time in her Dexta career that she had so directly referred to her Imperial connections. She had always been determined to succeed at Dexta without any help from Charles and without trading on her link to him. But DuBray, with or without the services of Erik Manko, was the most powerful obstacle she had encountered, and she had finally accepted the fact that she couldn't fight him with one hand tied behind her back. _Fight dirty,_ Chandra had told her.
"OSI has work to do in Quadrant 4," Gloria told him. "We'll be doing it on New Cambridge, during the Quad Meeting. Don't get in our way, DuBray."
DuBray considered her words in silence for a few moments, then nodded slightly. "Very well, then," he said. "You've decided on war."
"Call it what you want. I'm fighting back."
"You're being stupid, VanDeen. I'm not alone in this, you know. All four Quadrant Administrators will be aligned against you."
"From now on," Gloria said, "you should think of OSI as the Fifth Quadrant."
"THE FIFTH QUADRANT? BUT THAT DOESN'T EVEN make sense," Grant Enright protested. "You can only have four quadrants."
"You have no poetry in your soul, Grant," Gloria replied. "I kind of like it. And it exactly describes what OSI must become. In order to survive at all, we have to make ourselves as strong and independent as the Quadrants themselves, and their Administrators."
"Especially the Administrators," said Enright, frowning a little.
"It's not simply personal, Grant. I mean—yes, it's personal, but it's much more than that. The Quad Admins mean to destroy or neuter the OSI. They intend to make life as difficult as possible for everyone in this office." Gloria looked around the room, where the entire OSI staff had gathered. Some were sitting at the big conference table, and more were standing along the walls. All of them were staring at Gloria.
"We are at war for our survival, people," she said. "It's as simple as that. I think OSI is worth saving, and I'm going to fight for it." For a brief moment, Gloria thought she might be able to get away with turning this into a rah-rah, pep-rally speech, stirring the troops' blood and getting them into a fighting mood. But she thought better of it, and went in another direction instead.
"It's going to get nasty, and it's possible that sticking with OSI may not be good for the long-term health of your Dexta career. If anyone wants out, I'll arrange for your transfer immediately. I won't blame anyone who wants to make the sane, safe choice. You have a lot invested in being at Dexta, and I won't force you to risk it for my sake."
"Oh, Gloria, darling," Althea Dante interrupted, "don't be so melodramatic about it. Of _course_ we'll stay and help you fight the evil Quad Admins! I think it will be delicious."
Althea was probably telling the truth, Gloria reflected. Althea had some strange notions about what was fun, but Gloria was grateful to her for the strategic support. After Althea had spoken, no one else could think of anything they wanted to say. Gloria let the silence go on for only a few seconds before swiftly moving to change the subject.
"Elaine and I are off for New Cambridge in a Flyer tomorrow. But I really want to show the OSI flag at this meeting, so I'm sending a Cruiser, too. Jill, Althea, Brent, Darren, plus Arkady, and four or five of our Bugs. Our mission will be to assist Petra and Pug in their investigation on New Cambridge. Our _real_ mission will be to sell OSI to everyone at the Quad Meeting. We are going to work that meeting like a local politician working a town fair or a funeral. Althea, when you get there, I want you to plan an OSI reception and dinner some night. Spare no expense, and forget about the entertainment budget. Bill me, if necessary. Just make certain it is the one event that everyone will want to attend."
"Gloria, darling," Althea gushed, "I do believe this is the best assignment I've ever received at Dexta! I promise you, it will be a party to remember."
"Jill, Brent, and Darren," Gloria continued, "will join me at as many of the other receptions and parties as we can manage, and we'll haunt the committee meetings, general sessions, and the hotels, corridors, and restrooms. We are out to win friends for OSI, so we have to be friendly."
"Just how friendly?" Jill warily asked.
"I've never asked anyone in this office to screw someone for the sake of the job, and I never will. Anyway, we _can't_ screw everyone, so it's probably just as well if we don't screw anyone. We don't want to make any of them angry because they were left out."
"Gloria, I hope you are not trying to tell me that I shouldn't screw anyone on New Cambridge," Althea complained.
"As the Spirit moves you, Althea, as the Spirit moves you—as always. I'm just saying that it's not official OSI policy."
Gloria turned to look at Enright, the OSI Administrator. "Grant," she said, "I couldn't help noticing that it's crowded in here."
Enright shrugged. "It's the biggest conference room we have," he said.
"Get a bigger one," Gloria told him. "If we are going to be the Fifth Quadrant, we need to look the part. When I get back from New Cambridge, I expect OSI to have twice the office space it does now. Beg, barter, or bully, as necessary, but get it done."
"Yes, ma'am."
"And once we have all that new office space, we're going to need bodies to fill it. I want you to get us a dozen new Fifteens. Bring them in straight out of training and we'll raise 'em up right. Also, review all the transfer requests—I know we have a lot of them. See if you can snag half a dozen good Fourteens and maybe a couple of Thirteens."
"How are we going to pay for all those people? There's nothing in our budget for anything like that."
"Initially, we can pay them out of contingency funds. But you're right; we'll have to start submitting Resource Allocation Amendments. I know you've got friends in the Comptroller's Office. Wine and dine them, call in old favors, renew old threats. Whatever it takes."
Gloria turned from Enright and looked around the room at the rest of her people. "What I want," she said, "is for all of you to analyze the overall structure of Dexta. Look into some of the more obscure and neglected corners of the organization. Office of Weights and Measures, the Bureau of Reclamation, the Exo-Technology Review. Did you know that there is such a thing as the Dexta Ornithological Survey? I want you to identify all the bureaucratic orphans you can find and invite them to join the OSI family. We'll absorb some of their functions, promise to support them in their core missions, and offer them an ally in the never-ending battle against the tyranny of the Quadrants. There are a lot of offices in Dexta that are not under Quadrant jurisdiction. Consequently, they have little power in intramural squabbles and get rolled by the Quads at budget time. They're used to that and most of them don't even bother to fight back. But if we can get them organized and aligned with us and each other, they'll gain a lot of clout—at least, that's going to be our sales pitch."
"Hey, Gloria?" someone at the back of the room said. "It might help a lot if we could promise all these people some...uh...personal attention from the head of OSI."
"Good idea. Start scheduling lunches for after my return from New Cambridge. In fact, I think I'll throw a party at my penthouse and invite all our new allies and recruits. Drop some hints about that. Be sure to mention my deep, _personal_ gratitude to everyone who helps us. And it wouldn't hurt to remind them that I'm an Avatar of Joy now." She grinned at her troops. "Never can tell _what_ might happen with an Avatar of Joy!"
Grant Enright smiled along with everyone else, but shook his head. "I'm with you on this, Gloria," he said, "but I hope you realize what you're getting us into. This is nothing less than an attempt at an internal coup. A power play."
"Grant," Gloria said, "we don't have any choice. It's a fight we have to make. OSI either grows, or it dies. Think of it as our manifest destiny."
"Manifest destiny," Enright mused. "All right, then, manifest destiny it is. When do we invade Mexico?"
THE DEXTA OFFICE COMPLEX ON NEW CAMBRIDGE was a dark, massive structure built out of native rock. It dominated the skyline in downtown Central so thoroughly that the residents referred to it as "Gibraltar"—complete with a colony of baboons inhabiting it. The baboons, Petra and Pug discovered, were not friendly.
After a day of settling in and sightseeing, they approached the Dexta offices with a sense of eager anticipation. They reported to the Regional Office Administrator, or tried to, but soon found that no one was available to see them. It took them most of the morning to track down someone who could officially, if reluctantly, acknowledge their presence on New Cambridge.
"We want to examine old Quadrant records from fifty or sixty years ago," Petra explained to the administrator.
"Is that so?" said the administrator.
"Yes. Where can we work?"
"You'll have to find a spot."
"Can you recommend something?"
The administrator blandly shook his head. "Not really."
"Well, who can?" Pug asked.
"You'd need to talk to someone in Building Management."
But no one in Building Management seemed to want to talk to them. Another hour went by before someone's assistant deputy provided them with the necessary forms to be filled out. Forms completed, Petra and Pug waited expectantly to be assigned office space. They waited more than two hours. Finally, the assistant deputy's assistant told them to come back the next day.
The following day, after more hours of frustration, they were at last assigned an office. On closer inspection, the office turned out to be a utility closet. It was equipped with two desks, but no chairs, windows, or computers. Getting the computers took the rest of the day. They couldn't do anything about windows, but they did manage to steal two chairs.
On the third day they arrived early, ready to plunge into their assignment. It was a short plunge. They found that their computers could not access the necessary records. No explanation was immediately forthcoming. They spent most of the day finding someone who could (and would) speak to them on the subject.
"You want records from 3163?" asked a bored administrative coordinator.
"Yes!" cried Petra.
"Don't have 'em."
"What the hell do you mean you don't have them?" Pug exploded. "How could you not have them? The Quadrant Administration offices were right here in this building for over four hundred years. How could those records not exist?"
"Didn't say they didn't exist," the administrative coordinator responded. "Said we ain't got 'em."
"Well, who does?"
"Lessee...3163, you say?"
"Yes."
"That would be over in the Archives Section. Everything up to 3180 is there. That's when Mingus moved Quad Administration back to Earth, you know."
"So the records would be in the Archives Section's computers?" said Petra. "How do we tap into them?"
"You don't. Not from this building, anyway. Archives Section is a whole separate deal. Just down the block from here, in the Old Annex. Can't miss it."
Three days later, Petra and Pug at last settled in to begin their research in another utility closet in the Old Annex. They had uniformly been treated with opaque courtesy and bland indifference by the local Dexta staff, who regarded them not as plague-carriers, perhaps, but certainly as people who carried an unpleasant odor about them. For the first time, Petra understood at a visceral level why so many people throughout the Empire passionately hated Dexta.
She wanted to believe that Dexta got it right most of the time, and most of the time, it probably did. For all its flaws, shortcomings, overreaching, and outright idiocy, Dexta had somehow managed to keep the Empire humming for nearly seven hundred years. Dexta was the most successful bureaucracy in history, so it had to have been doing something right.
But successful or not, it was nevertheless still a bureaucracy, and dealing with it could be a nightmare for the citizens of the Empire. One way in which Dexta served the Empire was by functioning as a lightning rod for all the many resentments of the scattered masses, which would otherwise have focused on the Emperor. Whatever was wrong, it was Dexta's fault, not the Emperor's, who stood above the hurly-burly bureaucracy and thought only of what was best for his people. It was generally understood that Dexta thought only of what was best for Dexta.
The radicals of PAIN sought to exploit the popular displeasure with Dexta, but with little success, mainly owing to their anarchist agenda. Their ideological blinders seemed to prevent them from seeing that most people had little interest in overthrowing the established order. What they _really_ wanted was simply a validated tax stamp or an export waiver, processed and delivered with minimum fuss and bother. PAIN's recent campaign of attacks on Dexta facilities throughout Quad 4 (there had been two more incidents in the past week) had inspired no mass uprisings, only cries for better security.
Still, whatever their political shortcomings, PAIN operatives had at least succeeded in laying their hands on a trove of plasma rifles that had supposedly been destroyed fifty-five years earlier. Now that they finally had access to the Archive Section's computers (not to mention desks and chairs), Petra and Pug went to work with a will. Petra began tracking the rifles from their point of orgin in the munitions factories on Ostwelt, while Pug pored over the records of shippers, warehouses, and port facilities.
It soon became apparent that the Quadrant's data storage system had been designed for the convenience of contemporary bureaucrats, not for historical researchers in some nebulous future. Finding what they needed was neither quick nor easy. The computer could call up specific documents, but only by their titles and dates, which Petra and Pug didn't know. They could zero in on a general place, date, or bureaucratic cubbyhole, but after that they mainly had to inspect every document in a file to see if it was what they were looking for. Eventually, they accumulated enough information to give the computer more specific instructions, and things began to go faster.
The Mark IV plasma rifles, twenty-four thousand of them, were manufactured on Ostwelt by Thor's Forge, Inc., a division of Imperium Ltd. The shipment earmarked for Savoy was lifted to the Ostwelt Orbital Station on June 22, 3163, loaded aboard a Trans-Empire freighter, and departed for New Cambridge on June 25. The freighter arrived at New Cambridge Orbital Station on August 2 and unloaded the rifles in an orbital warehouse owned by Stavros & Sons, Inc. On August 24, the rifles were loaded aboard another freighter as part of a larger shipment of military supplies, and departed for Savoy on August 27. Since no records were available from Savoy, it was unknown whether the shipment ever actually arrived there. The freighter should have reached Savoy on September 4. The Ch'gnth attacked on September 8. The freighter was not heard from again and was presumed to have been lost in the Ch'gnth attack. An insurance claim for the missing freighter was filed by its owner, B & Q Shipping, Inc., in March 3165, and the claim was paid in full by Centron Assurance, Ltd., a subsidiary of Servitor, in December 3167.
"So we don't know for sure that the rifles ever reached Savoy," Pug pointed out that evening as they sat before a blazing fire in one of the industrial-sized stone hearths that populated the Ellison family home. "Maybe they didn't."
"Which would make sense," said Petra as she scrunched around a bit to get comfortable in Pug's embrace. "Presumably, if they reached Savoy, they would have been destroyed, along with everything else."
"But the freighter never came back," Pug said. "If it didn't go to Savoy, where _did_ it go, and why didn't it return?"
"There was a lot of other stuff in that shipment. Maybe if we knew what it was and where it came from, that might tell us something."
Pug nodded. "We'll need to check records from the warehouse, and lading bills from the shipper. So we'll have to track down B & Q Shipping and Stavros & Sons and go through their records."
"Also," Petra said, "we should get the Dexta documentation dealing with this shipment. Who put the shipment together, who authorized it, who accepted it at New Cambridge, who cleared transfer to Savoy? Fifty-five years isn't such a long time—some of the people who were involved with that shipment might still be around."
They fell silent for a while and listened to the crackling of the fire. Then Pug said, "I'll tell you what worries me about all of this."
"Me too."
"Me too, what?" Pug wanted to know.
"What worries you," said Petra. "Me too. It worries me too."
"But you don't even know what I was going to say."
Petra twisted around to get a better look at Pug. "Sure I do," she said.
"You mean it's gotten to the point where you can read my thoughts?"
Petra cackled like a villainess.
"Okay, Ms. Know-It-All, tell me what I was thinking."
"You were thinking that you were worried about the fact that there were twenty-four thousand rifles in that shipment, plus all that other stuff."
"Yeah," Pug said, "that was it, all right. If PAIN has some of the rifles, then maybe it has all of them. And the other stuff. They could have enough military hardware to equip a couple of infantry divisions. _That's_ what worries me."
"Me too," said Petra.
GLORIA EMERGED FROM DEXTA HQ ONTO THE teeming streets of Manhattan. It had rained earlier in the day, and the streets and sidewalks were still wet, but a fresh breeze from the north had blown the clouds away and the late-afternoon sky was a deep, clean blue. Shoppers and workers on their way home competed for navigation rights on the jammed sidewalks, and the high-pitched whine of a thousand skimmers contributed an air of urgency to the scene. Gloria sensed but scarcely noticed the ripple of attention that accompanied her as she made her way through the throng. Even in her raincoat and rain hat, pulled low over her brow, people recognized her. She was used to it; on a typical day, the three-block walk to her home might produce half a dozen requests for autographs, plus a smattering of smiles and friendly calls of "Hi, Gloria!"
She was well accustomed to the burdens of her celebrityhood by now and simply accepted the attention as a fact of life. On some days, she actively embraced it and paused to chat and pose for pictures with starstruck tourists. Today, however, she was preoccupied with more practical matters, like packing for her trip to New Cambridge and making certain that she had accomplished every chore that needed to be done before her departure. With Petra gone, she felt oddly vulnerable and disorganized.
She had walked less than a block when a young woman abruptly stepped in front of her. Gloria stopped short and stared at her. She was wearing a long black leather coat and had very short, spiky black hair and intense green eyes.
"Gloria VanDeen!" the woman cried.
Before Gloria could respond, the woman raised her right arm and extended it toward her. Clutched in her hand was a plasma pistol.
"Gloria VanDeen," she repeated. "In the name of the Peoples Anti-Imperialist Nexus, I hereby—"
Her speech was cut short by a swarm of Bugs. One of them stepped between Gloria and the woman, blocking Gloria's view. She didn't see exactly what happened, but a second later the woman was facedown on the pavement, with three Internal Security men piled on top of her. The plasma pistol skittered away from her.
Gloria instinctively stepped back, as did everyone else on the crowded sidewalk. The woman twisted and squirmed beneath the Bugs and tried to bite one of them on the finger. "Pigs!" she cried. "Fascist swine!"
The Bug who had stepped between them bent down and collected the weapon. He turned and looked at Gloria. "You all right, ma'am?" he asked shyly.
Gloria managed to nod.
The Bug was a very young man, fresh-faced and fuzzy-cheeked, with short blond hair and innocent blue eyes. Gloria remembered his name. "Thank you, Reynolds," she said.
Reynolds nodded, almost as if he were embarrassed. It occurred to Gloria that stepping between her and the woman with the pistol had been nothing less than an act of suicidal self-sacrifice, yet Reynolds had acted instinctively in a split second. Gloria also realized that, at such close range, a plasma burst would have burned straight through Reynolds and killed her, standing behind him. Reynolds must have known that. So his action had been purely symbolic; but that didn't make Gloria appreciate it any less. She was going to have to stop taking her retinue of Bugs for granted.
The three other Bugs manhandled the young woman to her feet. She spat at them, but missed.
"Better get back to HQ, ma'am," Reynolds said.
"I guess so," Gloria agreed.
SHE INSISTED THAT HER NAME WAS KRUPSKAYA, but her DNA said that it was actually Eloise Howell, of Greenwich, Connecticut. She was twenty years old, and she couldn't shut up.
"It doesn't matter that I failed. Others will succeed! VanDeen and all the rest of the Imperial oppressors will fall, like autumn leaves before the hurricane of justice! History demands it! The masses demand it, three trillion of them! PAIN is the revolutionary vanguard, and our sacrifice, our example, will inspire the masses to storm the ramparts of Imperial arrogance and bring down the fascist overlords and their running-dog lackeys. That means _you,_ Bug! Nothing you do can prevent it. Liberation is inevitable!"
"Pretty blabby, for an assassin," Arkady Volkonski observed. He and Gloria had been sitting before a view-screen in an observation room for an hour, watching and listening as Eloise Howell declaimed.
"Lucky for me," Gloria said. "If she had just pulled the trigger instead of stopping to make a speech..."
"I think the speech is what mattered most to her," Volkonski said. "Radicals live on rhetoric. What else have they got? It's more real to them than action. It is to her, anyway."
The Bugs had quickly hauled Eloise Howell back to Dexta HQ and turned her over to Counterintelligence. Three CI agents shared the small interrogation room with her and had taken turns browbeating, threatening, and tempting her with offers of leniency in exchange for information. That was purely routine. Eloise liked to talk, so they had let her talk. Eventually, she would run down and begin to realize that all the revolutionary rhetoric in the galaxy would not help her. At that point, they would begin administering the drugs that would wring her dry and extract every secret she knew, every name, every plan, every dream.
"What will happen to her?" Gloria asked.
Volkonski shrugged. "Dicenzo Four," he said. Gloria nodded. Dicenzo Four was where the Empire deposited its political prisoners, a world of high gravity and low expectations. Eloise Howell, late of Greenwich, Connecticut, would spend the rest of her life there.
"I almost feel sorry for her."
"Don't," counseled Volkonski.
"I know. But she's just a kid."
Volkonski raised his single, unpunctuated eyebrow. "How many middle-aged assassins have you seen? Booth was twenty-six, Oswald twenty-four, and Tancredi was only nineteen. Young, stupid, and idealistic."
"Lots of us were young, stupid, and idealistic once," Gloria pointed out. "But most of us didn't turn into assassins. Why did Eloise?"
"By the time our friends in CI finish with her, we'll know the answer to that. I'll see to it that you receive a summary. But don't expect anything profound. Her father probably forgot her birthday when she was eight, and that turned her to a life of radical activism. Or radical rhetoric, anyway. You want to know why Eloise stopped to make her speech? It's because she wasn't able to pull that trigger. She _wanted_ to be stopped. This way, she gets to make more speeches."
"Why would PAIN have chosen someone like her?"
"She probably chose herself. Or drew the short straw. My gut tells me that Eloise probably isn't connected to what happened on Cartago or the other terrorist attacks. She and her little cell of would-be terrorists probably heard about the other incidents and revved themselves up to strike their own blow for galactic liberation."
"You really think so?"
"Can't be sure at this stage," Volkonski said, "but I'd bet on it. For one thing, her pistol is a recent model, the kind you can get anywhere. No connection to the Savoy shipment. No reason to think this was part of some grand conspiracy. I doubt that we'll get anything useful from her."
"That's disappointing."
"It's typical. PAIN is not some sophisticated subversive cabal, Gloria. Until lately, they've never been more than a loud but minor irritant. People with a gripe against the Empire gravitate to it, but their anarchist ideology is self-limiting. How do you organize a bunch of people who reject the very concept of authority?" Volkonski shook his head. "PAIN has never posed a real threat to the Empire."
"But if they have the Savoy shipment...?"
Volkonski sighed heavily. "That," he said, "would require a reevaluation of the threat level. Somewhere, somebody a hell of a lot smarter and more dangerous than Eloise Howell must be calling the shots."
"Today's pig, tomorrow's bacon!" cried Eloise Howell.
Volkonski smiled slightly. "Oink," he said.
MRS. ELLISON HAD INVITED PETRA TO TEA. Petra equated "tea" with "coffee," and, therefore, with relaxed informality. Mrs. Ellison seemed to equate it with something entirely different. Petra sat next to her on the gold satin couch in what she assumed was the Tea Room, feeling uneasy about her jeans and tee shirt. Mrs. Ellison, as always, looked as if she were about to chair a meeting of the Well-Dressed Ladies Society.
They engaged in idle chitchat for a while. Mrs. Ellison was much better at it, since Petra had nothing to talk about except Dexta and Pug, and Mrs. Ellison's eyes tended to glaze over whenever Dexta was mentioned. That left Pug as their only point of common interest, and Petra sensed that Mrs. Ellison would not be pleased to hear about what a wonderful lover her son was.
Eventually, Mrs. Ellison's supply of idle chitchat was exhausted, and she fixed Petra with a cold, level gaze. "Ms. Nash, forgive me for being blunt, but may I ask, just what are your intentions regarding my son?"
"Intentions?" Petra asked, playing for time. "How do you mean?"
"Your intentions. Surely, that must be clear."
"I intend to stay with him, if that's what you mean."
"And if he were to leave Dexta and return here to assist my husband?"
"I don't know," Petra replied. "But I do know that Pug doesn't want to do that. He told me so, himself. He _likes_ Dexta, and he's proud of what he's accomplished there. It's the first thing he's ever done on his _own._ Do you know how much that means to him?"
"Better than you, my dear. I'm not unmindful of Palmer's desire for independence. Within limits, it's a healthy thing and I quite approve of it. Just as I approve of his relationship with you, in case you were wondering—also within limits."
"And those limits would be...?"
Mrs. Ellison frowned silently for a moment, as if deep in thought. Then she said, "Don't expect too much, Ms. Nash. You seem like a fine young woman, and I can understand my son's attachment to you. But his home and family are here—just as his future is here. If you expect to be part of that future, you must accept that fact. And you must learn what it means to be an Ellison."
"It means dressing for tea," Petra said. "I've learned that much."
Mrs. Ellison barely smiled. "That's a start," she said.
ELI OPATNU SAUNTERED INTO CORNELL DUBRAY'S office and found the Quadrant Administrator staring at the console on his desk. A million lights twinkled through the windows behind him. You could see a long way from the 110th floor.
"Someday I'm going to have an office like this," Opatnu said. "Maybe even _this_ office."
DuBray looked up and regarded the young Sector Administrator for a moment. "Everyone is in such a _hurry_ these days," he said with a hint of bemusement.
"Weren't you?" Opatnu asked.
DuBray leaned back in his chair. "As a matter of fact," he said, "no, I wasn't."
"Oh?"
"Fix us a couple of drinks," DuBray said, gesturing toward the bar to the left of his desk. "My usual." Opatnu did as he was instructed. DuBray got to his feet and stretched.
"When I was your age," DuBray told Opatnu, "we were in the middle of the war with the Ch'gnth. And the truth is that after those first, awful months, I had the time of my life. There I was, still a young man, paticipating in decisions that affected the fate of billions. It was heady stuff. I was almost sorry to see the war end. And, of course, I knew that when Norman moved up to Dexta Secretary—and the war guaranteed that he would—I'd take over Quadrant from him. So I really wasn't in a hurry. I didn't have to worry about my future."
Opatnu handed DuBray his drink, then took a sip of his own. "Must have been nice," he said, "not having to worry."
"But I get the impression that you _are_ worried."
"Not worried, exactly. Just say that I'm concerned. This Wendover business could still blow up in our faces."
"I doubt it," DuBray said with unforced confidence. "In-house investigations can always be controlled, one way or another. It doesn't matter whether it's the Comptroller or Internal Security, or even the newly minted Office of Strategic Intervention. It's simply a matter of knowing how to play the game."
Opatnu shook his head. "Maybe. But VanDeen won't be easy to control."
"I was under the impression that you had already accomplished that."
"One quick fuck isn't going to be enough. Not with Gloria."
"What's this? Are your powers slipping, my friend?"
Opatnu shook his head. "I have my enhancements; Gloria has hers. For Gloria, sex is _always_ great. She doesn't need me to find fulfillment."
DuBray considered this as he swirled the ice cubes around in his glass for a moment. "Perhaps so," he said. "But VanDeen is not the only woman in OSI."
"True," said Opatnu. "I've already got my eye on someone else who could be useful. But Gloria's people are remarkably dedicated. This isn't going to be easy."
"Easy or not, it must be done. You know what's at stake here."
"You mean, aside from my career—and yours?"
DuBray went to the bar and began fixing himself another drink. "My career," he said, "is perfectly secure. And yours, as well, if you don't take counsel from your fears. You're still young, Opatnu, so you lack a long-term perspective. Do you know how many investigations I've quashed over the years? Usually, all I have to do is rear up on my hind legs, let out a growl or two, and the problem is solved. VanDeen, I admit, presents an unusual challenge, but hardly an insurmountable one. If she continues to defy me and resist your persuasion, stronger means are available. I doubt that they will be necessary, but the option is always there."
"I heard that your 'stronger means' wound up in the hospital," Opatnu said with a slight smirk.
DuBray didn't quite snort. "Manko is useful, but he's hardly essential. I'm not talking about mere muscle here, or even sex. The purely physical is the lowest level of control. We use it because it's easy and it generally works. When it doesn't, we can turn to other methods. I can crush VanDeen anytime I want. I'd prefer to avoid that, but if she makes it necessary, I won't hesitate."
"I know," said Opatnu. "But sometimes, I wonder..." Opatnu trailed off and stared at the million glittering lights.
"What?"
Opatnu turned to look at DuBray. "If it's worth it," he said.
"It is," DuBray assured him. "But if you're having second thoughts about the choices you've made, I suggest that you get over it. When you finally get this office—and you will, when I move up to replace Mingus—remember that power is simply a tool, like a hammer. A carpenter doesn't sit around staring at his hammer, pondering its existential significance. He uses it to drive nails."
"I know," Opatnu said. He took a big swig of his drink. "And don't worry, I can handle VanDeen and OSI."
"Our friends expect no less," DuBray reminded him.
"Our friends," Opatnu said, "won't be disappointed."
THE DEXTA FLYER MANEUVERED AWAY FROM the dock at Earthport, accelerated smoothly for a few hours, then turned on its Ferguson Distortion Generators and slipped smoothly into its transluminal journey to New Cambridge, three-and-a-half days away. The Fergusons created a spherical membrane of eleven-dimensional Yao Space, which squirted through the inimical fabric of the universe, carrying within it a bubble of normal space with the Flyer at its center.
Travel between the stars was mainly a question of mass and energy. To save on both, a Flyer consisted of no more than a sealed cylindrical tube, twenty feet long and ten in diameter, with minimal facilities and life support for two passengers. No crew was necessary, so Gloria and Elaine were alone in the Flyer, which was the fastest available form of interstellar transport for human beings.
As soon as they were properly under way, Gloria stripped off her clothing, as she always did on such journeys. There was no point in using up four days' of laundry, especially when you could never be sure if there would be decent laundry facilities available at the other end of the trip. Elaine, making her first interstellar journey, followed Gloria's example.
Beds were built into each side of the cylinder, separated by a narrow aisle. There was a spartan, waterless bathroom aft. Two work consoles, chairs, a table, and galley facilities were crammed into the forward section of the craft. Gloria settled onto her bunk, took out her pad, and resumed reading her history of the war with the Ch'gnth.
Later, Elaine fixed their dinner—microwaved-fish-something, and a concoction that went by the name of DryWine; just add water, and you had a beverage that would have appalled any oenophile in the Empire. But it was palatable as long as you didn't imagine that it had anything to do with real wine, and it packed a considerable kick. By their third packet of the stuff, Gloria and Elaine were pleasantly high and chattering easily and freely.
Elaine's mother was a marine biologist, originally from Fiji, and her father was a Japanese businessman, currently serving a three-year term on a prison colony for fraud and embezzlement. She was philosophical about her family's disgrace, as she was about the fact that it had taken her three tries to pass the Dexta Entrance Examinations. She was Gloria's age, but seemed younger. Gloria knew all of this from Elaine's file, but it was interesting to hear about it from her own lips. Gloria had yet to figure her out to her own satisfaction, and hoped to get to know her better during their trip.
Petra, the Lap Dog, was friendly, smart, loyal, obedient, brave, helpful, efficient, and funny. Elaine, the Tiger-in-the-making, was clever, ambitious, tactful, wary, given to taking shortcuts, and a little too slick for her own good. Gloria liked her, but didn't really trust her, the way she did Petra.
Worse, there was an uncomfortable suggestion of hero-worship in Elaine's attitude toward her, and something more. It was as if Elaine believed that by observing Gloria closely, she might somehow learn and absorb her secret of success and make it part of herself. _Well,_ thought Gloria, _she's welcome to try._
"Another?" Elaine asked, indicating the DryWine.
Gloria hesitated, then said, "Why not?" There were enough packets of the stuff to keep them both blind drunk all the way to New Cambridge, which was probably the way some people preferred to travel. Elaine mixed the brew, poured for them, then lifted her glass in a toast.
"Here's to our first mission together," she said. "May it be successful, and not our last."
Gloria clinked glasses with her and took another gulp of the concoction. The first glass had been dreadful, but the fourth was merely bad. She had an idea about how to improve the fifth even more. Gloria got up, went back to her bunk, found her travel bag, rummaged around in it briefly, then returned forward carrying a small plastic bag.
She swilled down the rest of her glass, then handed it to Elaine. "Here, make another batch. Got an idea."
Elaine did as instructed. Gloria opened the plastic bag, took out a small handful of dried brown leaves, crushed them in her hand, then sprinkled the resulting powder into their glasses. She found a spoon and stirred vigorously until the powder had nearly disappeared.
"Jigli," Gloria explained. "It should improve this stuff, and it has some pleasant side effects."
"What is it?" Elaine asked. "I never heard of it."
"A little souvenir from Mynjhino. Drink up." They did. The tart, astringent taste of the jigli certainly didn't hurt the DryWine—as if anything could have.
_Why did I do that?_ Gloria wondered vacantly. She was getting pretty drunk, she realized, so maybe her judgment was a little cloudy. And maybe she just wanted to shut down her rational mind as completely as possible for the next few hours.
At war with the Quadrant Administrators. Spirit, how had she managed to bungle her way into such a fix? A Fifth Quadrant? Well, it sounded good, but the reality was that OSI was a tiny appendage of Dexta—a little toe, no more—while the Quadrants were the heart and lungs. Did she really believe that OSI could survive and triumph in such an unequal contest? Of course, she could always take the easy way out and go be Empress Gloria...
Elaine giggled irrelevantly. Gloria wanted to giggle irrelevantly, too. "What?" she asked.
"I was just thinking how great this is," Elaine answered, and giggled again.
Gloria found no cause for giggles. "If this is your idea of great," she said, waving her arm to encompass the cramped Flyer, "then you have led too sheltered a life."
"Not sheltered," Elaine said. "Definitely not sheltered. Just a little limited, maybe. But that's not what I meant. I meant that it's great, being alone here with you, getting plastered with Gloria VanDeen. You're my ideal, Gloria. I can't begin to tell you how much I admire and respect you."
"Don't, then."
"Really," Elaine persisted, "you've got the Empire by its balls, Gloria. You do what you want, no man owns you, and you have fun on your own terms. You do important work and save a lot of lives and make Dexta look good and...oh, just _everything_!"
Gloria sipped some more of the jigli-laced DryWine. The familiar jigli glow was spreading through her, warming her groin, tingling her limbs. And the DryWine was doing a job of its own on her slightly spinning head.
"So you wanna be like me?" Gloria asked Elaine.
"Of course!" Elaine gushed.
"Lemme tell ya, sometimes, _I_ don't wanna be like me. It's a full-time job, you know. And sometimes, people try to kill me. Ever smell your own hair being burned? I gotta be Gloria VanDeen allatime, everywhere I go. You think that's easy?"
"No, but it must be a lot of fun."
"Sometimes," Gloria said, "having fun can be a lot of work."
Gloria got unsteadily to her feet, made her way aft, and plopped down on her bunk. A moment later, Elaine sat down on the edge of the bed.
"Are you all right, Gloria?" she asked.
"Fine, just fine," Gloria assured her, then closed her eyes. Suddenly, she became aware of Elaine's lips pressing against her own.
She opened her eyes and tried to focus. "What are you doing?"
"Nothing I haven't done before," Elaine said, giggling again.
Gloria tried to sit up, but the slight weight of Elaine's body pressed her back down. "What the hell do you mean by that?" Gloria demanded.
"Remember that night, a couple of weeks ago, at the Club Twelve Twenty-Nine? I was there, Gloria."
"Spirit," Gloria groaned. She remembered the night, if not all the details. Not Elaine, certainly.
"It took a little doing," Elaine explained, "but I finally managed to squeeze in next to you. I kissed you, like this"—she demonstrated—"and then you kissed my tits. You seemed to enjoy it. Then I kissed yours"—again, she demonstrated—"and you seemed to enjoy that, too."
"Elaine—"
"So then I went on kissing you." Elaine shifted her weight and moved downward, twirling her tongue around inside Gloria's navel. Gloria shuddered.
"Elaine," she said, "maybe this isn't such a good idea."
Elaine looked up at her and said, "I think it is. Here, I have a treat." Elaine held her hand in front of Gloria's face. Two familiar lozenges were in her palm.
"Twenty-nine?" Gloria asked.
Elaine smiled and popped one of the lozenges into her mouth. The other, she placed between Gloria's lips, then continued her journey downward across Gloria's brown belly. She giggled again as Gloria's soft blond thatch tickled her chin.
_Not such a good idea at all,_ Gloria thought. And then she realized, as she shuddered again, that if she really wanted Elaine to stop, she should have said so much sooner. It was far too late for that now.
CLUTCHING PUG'S FOREARM, PETRA PAUSED AT the top of the short flight of polished marble steps that led downward into the vast, glittering ballroom. She took in a breath, took in the scene. Gliding gracefully around an area only slightly smaller than Weehawken, two or three hundred of the Ellisons' closest friends were dancing to the sedate rhythms of a large orchestra playing lush and lugubrious ballads from the twenty-fourth century. Everyone was beautiful or handsome, or both. Gemstones glinted and flashed incessantly as the stalwart men of New Cambridge maneuvered their extravagantly coiffed and garbed women around the dance floor. Liveried servants darted back and forth among the tables that ringed the room, while smiling bartenders kept a steady flow going. The dance number ended, the dancers politely applauded, then, almost as one person, turned to look toward the top of the stairs—at Petra.
It was one of those flash-frozen moments, and Petra knew, even as it was happening, that she would always remember it. The gentry of New Cambridge had come to this party in the expectation of seeing the young woman that the Ellison lad had brought home with him, and here she was—looking as good as she ever had in her life. Makeup just so, hair still brown but artfully highlighted and swirling down to her shoulders, dead on her ideal weight for once, thanks to all that Qatsima, and the dress was one she'd borrowed from Gloria and never given back. It was the dress she'd worn that night at the Governor's Reception on Mynjhino, and it left her mostly naked from her shoulders to far below her navel, with the dark blue smart fabric set at 70 percent transparency. The skirt was slashed in a wide vee almost to the crotch, and there was no back at all, to speak of. Ricky had loved it...
Poor Ricky. Dead, just days later. She hadn't worn the dress since then. Ricky had been the first great love of her life, blazing across her sky like a meteor, bright and hot and suddenly gone. Then there had been a brief fling with Bryce Denton, the sexy news weasel, but no one after that until she and Pug had found their way, after much travail, into each other's arms on Sylvania.
She loved Pug, although not quite the way she had loved Ricky; on the other hand, she hadn't loved Ricky the same way she loved Pug. Her love for Ricky had been crazy and fantastic, fierce in its intensity. With Pug the temperature was not quite as high, but she saw in him a love that might last.
So tonight was another necessary ritual, like Meeting the Parents. Now, she had to meet, not Pug's friends, since he scarcely knew most of these people, but his peers—the wealthy, powerful, and arrogant few who ran this planet. The social elite, the very apex of the food chain. She stared at them and they stared back, and for a moment the immense hall fell silent. Pug squeezed her hand, and together they started down the steps. Noise and chatter returned, and Petra had made her entrance. She resumed breathing.
Pug, who looked as if he had been born to wear a white dinner jacket, steered Petra through the throng with his palm on her bare back, smiling and nodding and making quick introductions. Petra felt as if she had been transported into one of those romance novels her mother constantly read, where life was nothing but a series of grand balls and swirling gaiety, punctuated by the occasional kidnapping by dashing rogues. There were even a few authentic Lords and Ladies present, well practiced in their charming condescension.
"Whoops," Pug said abruptly. "Moment of Truth approaching. Here comes one of my old girlfriends."
"Which one?" Petra demanded. "The one who wants to be a nun, or the one who died in the tragic Ferris wheel accident?"
"That one," Pug said, pointing toward a willowy blond with an intricate hairdo and heaving breasts the size of small melons, completely on display in an iridescent confection that had probably been woven together from the tailfeathers of some endangered species of hummingbird. Petra silently gulped, then quickly hit the contact switch in her dress seam, upping the transparency of her own gown to 80 percent. As Petra stepped forward to meet the competition, she found herself wondering how Gloria would have handled a moment like this.
"Pug, dearest!"
"Steff! You look wonderful tonight!" They exchanged quick kisses on their respective cheeks. Pug put his arm back around Petra. "Petra," he said, "I want to introduce you to an old and dear friend, Steffany Fairchild. Steff, this is Petra Nash."
They shook hands and smiled at one another. "Charmed," said Steffany. "We've all been looking forward to meeting you, Petra."
"All?"
"All of Pug's friends, I mean. He's been gone for over a year now, and we've been wondering about this mysterious new woman in his life. I mean, who would have thought that he'd find someone in Dexta?" Steffany laughed at the very mention of such an absurd notion.
"Petra," Pug said proudly, "is Gloria VanDeen's assistant."
"And he's mine," Petra added, possessively clamping an arm around Pug's waist.
"Yes," said Steffany, one eyebrow raised, "so I see."
"And what do you do, Steffany?" Petra asked, gamely trying to make conversation.
_"Do?"_
"Steff doesn't exactly _do_ anything, Pet," Pug explained.
"Now that's not true, and you know it. I'm on half a dozen charity committees. Why, just last week your mother and I spent a whole afternoon together, trying to come up with a theme for this year's Founders Day Ball."
"A whole afternoon," said Petra trying to sound impressed. "And what did you settle on?"
"We were torn between 'A Midsummer Night's Dream' and the Sultan's Seraglio. The full committee will have to decide."
An old school chum suddenly appeared and dragged Pug away, leaving the two women staring at each other in brittle silence. Finally, Steffany said, "Charming dress, Petra. But perhaps a little too 'New York,' if you take my meaning. Here on New Cambridge, we keep our pussies covered."
"Yes," said Petra, "I'd bet money on it."
Steffany crossed her arms and tilted her head to one side. "Think you can keep him?"
"I haven't seen any reason yet to think that I can't," Petra replied.
"Really? Then perhaps you should look around you. If you think I'm your only competition for Pug, you're either naïve or foolish. The Ellisons are not about to let their son spend his life as a file clerk—or as a file clerk's assistant. You may think those cute little tits of yours will keep him hooked, but you're dealing with primal forces here, Petra. You might keep that in mind."
A little later, Petra met some of the primal forces, face-to-face. Pug steered Petra over to a knot of distinguished-looking people and made the appropriate introductions. "And this," he concluded, "is my great-uncle Benedict, my mother's uncle." Uncle Benedict had silver hair and piercing blue eyes that took Petra in from head to toe with evident approval.
"Congratulations on the new appointment," Pug said. He turned to Petra and explained, "Uncle Benedict was just named Imperial Governor of Pelham III, out on the Frontier in Sector 23."
"Yes," said Benedict, "a thousand and twelve dreary light-years from Earth, I'm afraid, but an interesting position, nonetheless. My first go at managing indigs."
"Indigs" was Empire-speak for indigenous populations, meaning alien civilizations. Petra nodded and asked, "Have you governed elsewhere?"
"Third time around," Benedict said. "Once for Darius, once for Gregory, and now, Charles has called upon me. Wasn't really eager for the assignment, but I was at loose ends anyway, and one doesn't say no to an Emperor, does one?"
"I never have," Petra agreed.
"As a matter of fact," said Benedict, "I wanted to have a talk with you about this, Palmer. No time like the present, I suppose. Fact is, I have a few slots to fill in the Dexta portion of my staff. Think you might be interested in taking a crack at being my Undersecretary for Administration?"
Pug's eyes widened. "Are you serious, Uncle Benedict?"
"Entirely. Undersecs are Thirteens, of course. I gather you're still a Fourteen?"
Pug nodded.
"Figures," said Benedict. "Promotion in Dexta is so damnably slow."
"Not really," Pug said. "I mean, I was a Fifteen for less than a year, and I got the bump to Fourteen just last summer."
"Still, for a young man of your quality, it's such a waste to linger in the lower echelons. That's why I want you for this Undersec slot, my boy. My Imperial Secretary will be a carryover, and I'll need her experience for the first year, of course. But after that, I can request her transfer and promote you to fill the position. A year from now, you'll be an Imperial Secretary and a Twelve. What say?"
Pug didn't quite gape or dither, but he was clearly taken aback by the offer. He glanced uncertainly at Petra, then back at his great-uncle. "I don't quite know _what_ to say, Uncle Benedict. I thank you for the offer. It sounds like a wonderful opportunity."
"I should think so," said Benedict. "Yet I sense hesitation." He raised an eyebrow and peered expectantly at his young relation.
"Well...it's kind of...uh, sudden. And, uh, Petra and I..."
"Ah, I understand. Ms. Nash, you are, I believe, a Thirteen?"
"That's right."
"Well, I believe I might be able to open up another Undersec slot on Pelham, come to that. A Dexta position, of course, so I'd have to finagle a bit. They don't seem to appreciate Imperial appointees messing about in their cricket pitch. Might take a few months to clear it, but I see no reason why you couldn't join Palmer in relatively short order."
"Uh...that's very considerate and generous of you, sir," Petra said.
"Then it's settled?"
Pug and Petra looked at each other for a moment. Then Pug looked back at his great-uncle and said, "I certainly appreciate your offer, Uncle Benedict. We both do. It's just that we already have a commitment to our present positions in the OSI, and I don't know if we could just walk away from that."
"Don't be silly, lad. This is a Spirit-sent opportunity for a capable youngster such as yourself. I know your parents would be very disappointed if you were to turn your back on this. And as for OSI"—Benedict cleared his throat—"I have it on good authority that Ms. VanDeen's rogue operation is about to be consigned to the ash heap of Dexta's history. You'll do yourself no good, my boy, by clinging to the timbers of a sinking ship."
"Sinking ship?" Petra protested. "Now just a—"
"What Petra means," Pug quickly interjected, "is that there are personal and professional loyalties involved. We can't simply turn our backs on Gloria."
"Do you honestly believe that she wouldn't do the same to you, if it suited her purposes? My sources at Court—and they are very good sources, at that—tell me that Charles has made a very attractive offer to Ms. VanDeen, and that she is seriously considering returning to him."
"What?" cried Petra. "She'd _never_ do that. Never!"
Benedict smiled at her indulgently. "I'm sure you want to believe that, my dear," he said, "but those 'personal and professional loyalties' you seem to value so much are entirely ephemeral, let me assure you. You're still very young, so you believe that everything is eternal. When you're older, you'll realize that nothing lasts, nothing endures—certainly not anything as insubstantial and abstract as 'loyalty.' "
"Gloria is my best friend," Petra insisted.
"Is she, now? Well, then, perhaps she'll take you along when she returns to Charles. Personal secretary, that sort of thing. But if you intend to remain with Dexta, come what may, then I think you should consider the position I've just mentioned. And you, as well, Palmer. I don't require an answer this evening, in any event. I'll be around for the Quadrant Meeting. We'll talk again." With that, Benedict turned away from Pug and Petra and struck up another conversation.
Petra eyed Pug uncertainly as they walked away. "Pug?"
Pug pulled her a little closer. "Don't worry about it," he told her.
But she did.
WORRIES NOTWITHSTANDING, IT WAS AS GLAMOROUS and enjoyable an evening as Petra could recall. She whirled around the dance floor with Pug, feeling lighter than air and as bubbly as champagne. She knew she was the center of attention and that many eyes were upon her as she spun and twirled to the music. Throwing caution to the wind, she set her gown's transparency at 90 percent and let her shoulder straps fall, flouncing around the room nearly as nude as she had been during her stint as a bar girl in Elba's Emporium back on Sylvania. She was determined to let the nobs and snobs of New Cambridge see that the Ellison lad's new flame was unafraid and unintimidated. She wondered if Gloria felt this way all the time.
In due course, she met two more of Pug's old girlfriends, neither of whom seemed to be a candidate for a convent or a mortuary. She also met, and danced with, a gaggle of his old pals. One of them pawed her breasts, squeezed her mostly bare bottom, and invited her to join him in one of the upstairs bedrooms. She received similar invitations from one of Pug's uncles and a cousin. She politely declined them all, but began to understand why Uncle Benedict thought that loyalty was merely a transitory abstraction.
Late in the evening, Pug led Petra into one of the anterooms and introduced her to a woman he identified as "Aunt Saffron," even though she was not really a relation but, rather, an old and close friend of his mother. She was a tall, thin woman with exquisitely styled blond hair and narrow, aristocratic features that seemed vaguely familiar to Petra.
"You're looking well, Palmer," she said. "And from what I hear, it seems that you've done well in that... _place._ You'd do well anywhere, of course, but I still wish that you'd taken my advice."
"Now, now, Aunt Saffron," Pug soothed, "don't get started. I know how you feel about Dexta, but I've made my decision and I have no regrets."
"Everyone starts out with no regrets. They come with age and experience, and I'm confident that you'll collect your share of them. I certainly have." She looked toward Petra. "And you, Ms. Nash, no regrets either, I suppose?"
She snuggled a little closer to Pug and squeezed his hand. "Not tonight," she said.
"I presume you'll be going to Pelham together, then? Well, that's all to the good. The farther away you get from Manhattan, the better off you'll be."
"We haven't made up our minds yet," Pug said. "I see that Uncle Benedict has already told the family about his offer."
Saffron assayed a rather grim smile. "More the other way around, I think," she said. "I gather that your mother discussed the matter with him."
Pug nodded. "I figured as much," he said. "I wish they'd just stay out of it and let me do things for myself."
"A wise son follows his parents' advice."
"Ain't that the truth?" said a ruddy-faced man who had suddenly appeared at Saffron's side. He wrapped an arm around her and leaned over to give her a kiss on the cheek. "Every word of advice you ever gave me is engraved on my heart. 'Stick around until the old man croaks,' you said, 'and all of this will be yours.' So I did, and it is."
"I never said any such thing."
"You didn't? I could have sworn that was you. Maybe it was Nanny, or the gardener. How's it going, Pug?"
Pug extended an arm and shook hands with the man. "Good to see you, Whit. Petra, this is Saffron's son, Whitney Bartholemew, Jr. Whit, say hello to Petra Nash."
"Hello, Petra Nash," he said, taking Petra's hand and kissing it gently. There was alcohol on his breath, and from the way he swayed back and forth ever so slightly, it was apparent that he'd had a lot of it this evening. He looked to be in his forties, perhaps, and had black hair, a high forehead, and sharp, dark eyes.
"So," Bartholemew said, "are all the women in Dexta as sexy and beautiful as you, or did Pug just get the pick of the litter?"
"Random drawing," Petra explained.
"Ah, it was luck, then. Should have realized. The Ellisons have always been big on luck, you know. While we Bartholemews have always relied on our native cunning and rapacity, Mother's side of the family, on the other hand, ran toward meticulous plotting and cruel calculation. So here I am, the offspring of buccaneers and bureaucrats. I've always pitied young Pugnacious, here, bound as he is by ten generations of stiff-necked rectitude. As a Bartholemew, I have a little more latitude to be a rebel, you see."
"Some rebel," Pug snorted. "He runs a dozen companies and never gets out from behind his desk."
"Not true! I get around more than you think, my boy. I might just surprise you someday." He spun slowly around and pointed toward the glittering throng. "Surprise all of these bastards, just see if I don't."
"Why don't you surprise your mother, then," Saffron said, "and see if you can get through this evening without making yet another scene?"
Bartholemew leaned close to Petra and said in a stage whisper, "My mum doesn't trust me to behave myself."
"Neither does mine," Petra said with a conspiratorial smile.
"Ah! We are kindred spirits, then! Come away with me, Petra Nash, and help me make a scene."
Before Petra could reply, Pug put his arm around her and pulled her away. "I saw her first," he said. "Great to see you, Aunt Saffron. Take care, Whit." Petra waved a weak farewell as Pug hustled her away.
"Whit's a bit loaded, as usual," Pug told her. "A good guy, but he has a tendency to get carried away. Since his father died a couple of years ago, Saffron has been trying to get him to grow up, finally."
"She doesn't appear to have succeeded."
"Oh, Whit's not so bad. He just hasn't found himself yet. His family situation was...well, it was difficult."
"I see," said Petra. "What about his mother? What's her story? I mean, what does she have against Dexta?"
"Everything," Pug said.
"Come again?"
"Her full name, in case I didn't mention it, is Saffron Mingus Bartholemew."
"Mingus?" Petra's eyes widened. "Not...?"
Pug nodded. "She is the daughter of Norman Mingus. And," he added, "she hates him."
GLORIA SETTLED INTO HER SUITE IN THE Imperial Cantabragian, the ritziest hotel on the planet, feeling drained from the boredom of her trip and the hassles of securing lodging. With the Quadrant Meeting about to begin, every hotel room in Central seemed to have been booked a year or more in advance. Gloria, with fame and money to spare, had been able to prevail upon the manager of the Cantabragian to find an appropriate suite, which required a difficult and diplomatically touchy reshuffling of various Dukes, Governors, and single-digit Dexta nabobs. Being who and what she was could be a strain at times, but it could also open doors—literally in this case—that would have remained closed to anyone else.
Elaine, who had a bedroom of her own in the suite, was off on a mission to the local Dexta offices to arrange for security. The impending Quadrant Meeting had taxed local resources to the limit, but, however reluctantly, IntSec would be forced to provide a round-the-clock contingent of Bugs to safeguard Gloria VanDeen. It wouldn't be wise to let harm come to the most famous woman in the Empire on their home turf.
"I just got here," Gloria said to herself, "and I've already inconvenienced scores of people. No wonder everybody hates me." Well, not everybody—just the Quad Admins and everyone who valued their future at Dexta. Spirit, life was complicated!
It had become more complicated than necessary during the Flyer voyage to New Cambridge. After that first night, Gloria had explained to Elaine that she never had sex with anyone who worked for her at OSI, and that what had happened between them could never happen again. Rather than being disappointed by this news, Elaine had seemed, somehow, to be buoyed by it. She had scored with Gloria VanDeen, after all, and that made her unique within OSI.
It had been a stupid thing to do, Gloria readily conceded. Pleasant and satisfying, but stupid nonetheless. She knew Elaine would crow about her conquest, and that could cause problems with others, whom she had already turned down. She'd probably have to transfer Elaine and get her out of OSI, preferably off Earth. Punish the ambitious little tart for the crime of having had sex with the great and powerful Gloria VanDeen, Avatar of Joy.
Yet Elaine had done nothing that Gloria hadn't done herself—sleeping with a powerful superior to secure her position and advancement at Dexta. The game never changed. Cornell DuBray from above and behind, Elaine Murakami from below; she had been well and truly fucked by both of them, and she could hardly claim that it had been done without her consent. Dexta was becoming indistinguishable from the null-rooms she frequented. She floated at the center of the swarm, seen, desired, and targeted by one and all.
If only Elaine hadn't given her the Twenty-nine, maybe she would have been able to avoid what happened. But, after a moment of stunned hesitation, Gloria had sucked the lozenge into her mouth. Seconds later, she began to feel the rapturous intensity of the drug as Elaine industriously stoked her fires even higher. But the jigli had been her own idea. Why had she done that? At some level, had she welcomed and encouraged Elaine's advance?
And why not? Why not keep Elaine around? What was so bad about having a small, sleek body at her beck and call, as a change from all those big, hairy ones? She knew that other successful Dexta Tigers often kept small-to-medium-sized harems of eager and devoted young men and women; why shouldn't she? If she willingly shouldered the aching burden of responsibility, shouldn't she take full advantage of the perks that came with the job?
"Gotta get a grip, girl," she said aloud.
GLORIA WAS SO GLAD TO SEE PETRA THAT SHE gave her a big hug as she entered the suite. "Missed you, kiddo," she said.
"Missed you too, Your Avatarness. Gloria, we're going to need some of your star power if we're ever going to get anything done on this planet." Petra was looking more Tigerish than ever, Gloria noted, in a minimal miniskirt and sheer lime-green silk shirt, provocatively unbuttoned to her waist. And she had a bit of a glow about her, as if she were radiating energy in some forbidden wavelength.
"New Cambridge seems to have agreed with you, in spite of everything," Gloria observed. Petra didn't quite blush.
"Gloria, I swear I feel like one of those girls in the Greek myths who get swept off their feet by some handsome young god. Only this is the part where he brings her home to Olympus to meet the family and friends. They had this huge party, everyone dripping with money and ego...and Gloria!...I was the belle of the ball! I mean, really, I was! Because I was the mysterious woman in Pug Ellison's life, you see. They were all looking at me, and I was looking right back at them, and you should have seen me! I looked so good, I looked like _you,_ sort of. I mean, I was the most glamorous Petra Nash that ever was, and I loved every second of it! Spirit, Gloria, is that the way you feel all the time?"
Gloria couldn't help laughing at her friend. She put her arm around Petra's back and guided her to a sofa, where they sat down.
"I only feel that way some of the time, alas. And I'm happy for you that you got a chance to feel that way. Fun, isn't it?"
"Gloria, you have no idea—I mean, wait a minute— _you_ have an idea. But me! I'm just little ol' Petra from Weehawken, and suddenly there I was, feeling like Eliza Doolittle at the ball. I was the absolute center of attention, and I swear, I handled it like a pro. I was witty and charming and sexy, and I didn't spill a single drink or get caviar in my hair or step on any toes but Pug's. And he didn't mind." Petra smiled.
"And where is the Pugnacious One?"
"Oh, he got shanghaied into some family thing today, but you were my excuse to avoid it."
"Problems with the Ellisons?"
"Not really problems," Petra said. "They've all been friendly and gracious and everything. But it gets a little overwhelming sometimes. I mean, you're the only rich person I know, and you're almost normal, most of the time. But the Ellisons! Gloria, every time I turn around, I bump into some servant determined to be helpful. Or see some mural or tapestry depicting the glorious heritage of the Ellisons. Yesterday, Pug's mother asked me about my debut. My debut! At first, I didn't even know what the hell she was talking about."
"She probably doesn't encounter many young women who _haven't_ had a debut," Gloria suggested. "How's the work going? You may recall, this _was_ a business trip."
Petra frowned. "Pug and I are being treated like lepers."
"Get used to it," Gloria told her. "We are now more or less at war with the Quad Admins, which means, effectively, the rest of Dexta." Gloria brought Petra up-to-date on recent developments at Dexta HQ. She mentioned that she had dealt with the Manko problem but didn't say how. Petra then treated Gloria to an account of her adventures with the New Cambridge office.
"Anyway," Petra concluded, "we've made some progress, but we still have to do a lot more digging. We haven't even started with Stavros & Sons and B & Q Shipping, but we have managed to find a slew of Dexta material related to that Savoy shipment. We even identified a few Dexta people from that time who are still active, but we haven't been able to meet with them face-to-face yet. But Gloria, we did learn something that's pretty interesting, especially in light of current events. The guy who was the Assistant Quad Admin in 3163, and who signed off on the Savoy shipment? It was Cornell DuBray!"
Gloria nodded. "I knew he was Mingus's assistant at the time, but I didn't realize he had anything to do with the Savoy shipment. I assumed it was all handled routinely at a lower level."
Petra shook her head. "From what I've seen so far, there was nothing routine about that shipment. It seemed to generate more than the usual amount of paperwork, and people at the upper levels were definitely involved with it. I've even seen a couple of memos on the subject that were signed by Norman Mingus."
"When I talked to him about it," Gloria said, "he told me that there was so much going on at the time that he didn't really have any specific memories of that last shipment. But it figures that he would have dealt with it. I just finished reading a history of the war, and apparently the last few weeks before hostilities broke out were a real madhouse, especially on New Cambridge. You know, Savoy is only about seventy-five light-years from here, so this planet was practically on the front lines. Mingus had to make a lot of important decisions on his own, without any direct oversight from Earth. That's one reason why after the war, when he became the Secretary, he pulled all the Quad Admins back to HQ. He thought Quad Admins shouldn't have the kind of independent power that he had when he was here."
"Speaking of Norman Mingus," Petra said, "you'll never guess who I met. His daughter! Saffron Mingus Bartholemew."
"No kidding!" Gloria knew that Mingus had a raft of children from his five marriages; she had even gone to summer camp with one of them when she was eight. But Mingus never discussed his personal past, and Gloria had been unaware of Saffron's existence.
"You can sort of see the resemblance, once you know," Petra said. "She's probably in her eighties and still looks great. Pug says she grew up here on New Cambridge when Mingus was Quad Admin. She was a buddy of Pug's mom in school. These days, she seems to be a wealthy widow with a wayward son. Pug says she hates her father."
"Really? Norman's coming to the Quad Meeting. I wonder if they'll get together? I know he's estranged from some of his family; but it's such a big family, and with five marriages, I suppose something like that is inevitable. Still, it's kind of sad. He's so isolated. I don't think he really has anything in his life now except Dexta."
"And you?" Petra ventured.
Gloria thought about it for a moment, then shook her head. "I think he views me as a harmless indulgence of his old age, but he won't let it go beyond that. I don't think he's truly very close to anyone, and probably prefers it that way. After the life he's led, I suppose that's not surprising. Petra, until I read that book, I don't think I really appreciated what a great man Norman Mingus is! We might not have won that war if it hadn't been for his leadership in this Quadrant. He's probably one of the four or five most important men in the history of the Empire."
"And his daughter hates him," Petra added.
"I suppose," said Gloria, "that it's hard to pay the proper amount of attention to your family when you've got an empire to run."
"Yeah," said Petra. "Family and career." She looked pensive for a moment, then turned and stared Gloria in the eye. "Gloria? Are you going back to Charles?"
"Spirit! Does _everyone_ know about that?"
"Pug's great-uncle Benedict knew. He's an Imperial Governor out on the Frontier, and he wants Pug to come work for him. First as an Undersec and a Thirteen, and then within a year he'll move up to ImpSec and a Twelve. He also said that he might be able to find another Undersec slot for me."
"I see."
"Pug hasn't decided. I mean, I think he wants to stay with OSI, but I know there will be a lot of pressure on him from his family. And, well...if you're going back to Charles..."
"Petra, I honestly don't know what I'm going to do. He offered me real power as Empress, and with everything that's happening with the Quad Admins, I admit it's tempting. But for now, I intend to fight it out at OSI. I hope you and Pug will want to stay. But I understand the situation. Whatever you decide, you know you have my blessing."
"And you have mine, Gloria," Petra said.
THE SKIMMER TAXI DEPOSITED PETRA AT THE specified address in the Old City, near the harbor. After her visit to Gloria, she had decided to spend the rest of the afternoon checking out B & Q Shipping. But now that she was here, she wondered if maybe she should have waited until Pug was available. The neighborhood was seedy, the buildings were in disrepair, and the people watched her with avaricious interest.
Gusty winds from the harbor did dangerous things with her brief miniskirt and her unbuttoned shirt. Being nearly naked at the ball or at Elba's was one thing, but it felt risky to be parading her bare breasts and bottom in a place like this. And yet...there was something oddly delicious about it, too, and she realized that risk was part of what made it fun. Anyway, she knew some Qatsima now. She had even thrown Gloria! Petra smiled at the dangerous-looking denizens of the street as she passed them, and made no attempt to cover herself.
A faded sign on the exterior of the brick building told Petra that she was at the right location. She entered the front door and found herself in a dusty, mostly deserted complex of glassed-in office space. Glancing at a sign, she learned that B & Q Shipping was on the top floor; another sign informed her that the elevator was not in service. With a weary sigh, Petra assaulted the five flights of stairs. The surface gravity of New Cambridge was 1.06 G, and the difference was enough to give her sore feet.
Panting slightly, she arrived at the top floor and saw that here, most of the office space was unlit, and the only sign of life was an old man with ragged gray hair sitting at a console, staring at images of nearly naked women. Petra discreetly cleared her throat, and the old man looked up abruptly. His alarm quickly turned to delight as he focused on the living, breathing, nearly naked woman standing before him.
"Well, hello there, cutie," he said, breaking into a grin. He was a centenarian, at least, Petra figured, maybe even 120. It was hard to tell because at around the century mark, the effectiveness of the antigerontologicals began to decline, at rates that varied from individual to individual.
"Good afternoon," Petra said, smiling. "Is this B & Q Shipping?"
"You're lookin' at it, sweetie pie. I'm the 'Q.' Jamie Quincannon." He pushed himself up to his feet and held out his hand. Petra took it in hers.
"I'm Petra Nash, from Dexta, and I'd like to talk to you, Mr. Quincannon."
Quincannon's left eyebrow rose slightly. "Dexta, you say? Yeah, I heard they were havin' their big convention here. You lookin' for a room, sweetie? Do a little real estate on the side, y'see, and I just happen to know where I could get you a furnished apartment for the length of the convention. Prime location, just up the coast in Overlook. Only fifteen minutes from the Transit."
"Uh, thank you, Mr. Quincannon, but I already have accommodations." She suppressed an urge to point out the window at the Ellison compound looming high on the cliffside above Central. "What I need is to talk with you, or someone, about a shipment B & Q made about fifty years ago. I'd also like to look at your records."
"My records? Y'got a warrant?"
"No. I could get one," Petra told him, "but you don't want to make me go get a warrant, do you?" She smiled again and leaned forward a bit to give Quincannon a better view of her small, round breasts.
Quincannon noticed and smiled back at her. "Naw," he said. "Say, you bein' from Dexta, would you know if that Gloria VanDeen is comin' to the convention?"
"She's already here," Petra said, her smile taking on a slightly rueful complexion. Even here, in the midst of vamping a dirty old man, she was overshadowed by Gloria. Quincannon proceeded to emphasize the point by turning back to his console and tapping a few keys to bring up an image of a very naked and stunningly beautiful Gloria.
"Damn, she sure is somethin'! Hope I get to see her while she's in town. You wouldn't happen to know her, would you, sweetie?"
"Would you believe me if I told you I was her assistant?"
Quincannon's eyes widened. "You wouldn't be pullin' an old man's leg, would you, cupcake?"
"Not a bit of it. In fact, I'm here to work on an assignment for Gloria. We need to learn all we can about a B & Q shipment in September 3163. It was the last shipment to Savoy before the start of the war. Were you here then, Mr. Quincannon?"
"Told you I was the 'Q,' didn't I? Sure, I was here. Remember the shipment, too. Why do you want to know about it all of a sudden?"
"Mr. Quincannon, weapons from that shipment have turned up in the hands of terrorists. We need to find out what really happened to that shipment. We don't think it ever reached Savoy."
Quincannon took a heavy, rasping breath, almost a sigh. "Terrorists, you say?" He shook his head slowly, like a judge confirming a sentence. "Told Bart. Told him. Knew it would come back to bite us. But did that stiff-necked son of a bitch listen? Did he ever?"
"Bart?"
"My late partner. The 'B.' Died three years ago. Now there's only me here, give or take his son. Junior's got other fish to fry, but he keeps the office open. When he needs a special shipment for one of his other businesses, he leases a freighter through B & Q, which means that I lease one and pass it on to Junior at twenty percent under cost. Tax deal. Junior makes more money when B & Q takes a loss, y'see. But it gives me somethin' to do once in a while. Y'know, cutie, we used to own— _own_!—twenty-three freighters. Whole building used to be B & Q offices. Now, we got one office, one employee, and no freighters. Anyway, that whole shipment was Bart's deal. Didn't want to touch it, myself."
"The insurance company paid you for your missing freighter, though," Petra reminded him.
Quincannon grinned suddenly at the memory, then resumed his usual slightly suspicious countenance. "Yeah," he said, "they did. Bart's deal again. Knew how to play both ends against the middle, my partner did."
"And what actually happened to that freighter, Mr. Quincannon? It didn't go to Savoy. Where _did_ it go?"
Quincannon shook his head with a bit of vigor for emphasis. "Told ya, it was Bart's deal. Far as I know, that freighter got blown to shit by the Ch'gnth."
"I see," said Petra, frowning. "In any case, I'm going to have to look at your records. We're particularly interested in getting a cargo manifest."
Quincannon pointed to another console on the other side of the room. "Help yourself, honeybunch. Bart's dead, and I don't give a rat's ass, so go ahead. Download any damn thing ya want. Just don't tell the kid I let you do it, okay?"
"The kid?"
"Bart's son. Whitney Bartholemew, Junior."
"Oh," said Petra. " _That_ kid."
GLORIA, PETRA, JILL CLYMER, ALTHEA DANTE, Elaine Murakami, and Brent Rostov and Darren Mogulu, two of OSI's bright young men, huddled in a corner of the immense room where the reception was being held following the opening session of the Quadrant Meeting. More than five thousand Dexta representatives and interested onlookers from throughout the Quadrant, having endured three hours of speeches and platitudes at the famed Central Opera House, were now jammed onto the floor of the city's main sports arena, where food, drink, and important people were freely available. Norman Mingus had not yet arrived on New Cambridge, but Cornell DuBray was somewhere in the room, accepting the obsequies of the toiling Dexta masses who worked for him.
"Try to avoid DuBray," Gloria instructed. "I don't want any big, ugly scenes—at least, not this early in the conference. Plenty of time for that. Meanwhile, keep circulating around your assigned area and schmooze every breathing soul, except the waiters. I don't want _anyone_ to leave this conference without having had face time with someone from OSI. I want us to be _noticed_!"
There was little doubt that they would be, since among them, the five women wore enough fabric to decently clothe no more than two or three of them. Althea Dante, the petite, alabaster-skinned, raven-haired Imperial Coordinator, whose reputation within Dexta was even more notorious than Gloria's, wore nothing at all except for some oversized gems depending from golden chains around her neck and hips. The other women were at least partially covered, if not exactly concealed. For her part, Gloria wore a swooping, deep purple toga-dress, held together by diamond brooches at her right shoulder and left hipbone, leaving her left breast, leg, and buttock entirely bare. Brent and Darren, meanwhile, were decked out in their Imperial finest, as handsome as the women were beautiful.
"And remember our message," Gloria said. "OSI is here to help one and all, to be an advocate for the bureaucratically disadvantaged. Our one and only goal in life is to make Dexta function like a well-oiled machine for the benefit of all concerned. We are a threat to no one and a friend to all."
"And if that doesn't work," Althea added, "I've staked out a utility room over there where you can screw their brains out."
"Purely optional," Gloria assured the others. "Everyone got their pin-pads?" Pin-pads were tiny receivers linked to their personal computer pads, and each woman was equipped with one. "Make plenty of notes—in fact, do it after every conversation, if you can. We are going to follow up each personal contact with a letter that will make reference to your conversation, so be sure to record some specifics. The personal touch is vital. Okay, ladies and gentlemen, go do your stuff."
Petra, Jill, Elaine, Brent, and Darren turned and walked off to their designated quadrants of the Quadrant, while Althea lingered to discuss some details concerning the OSI reception. She had already handed out a small fortune in bribes to cooks, caterers, musicians, staff, and managers in order to secure a workforce for the date she had chosen. But finding a suitable venue not already booked proved to be a problem that resisted her largesse.
"Gloria," she said, "it's hopeless, unless you want to rent an alfalfa field and put up a circus tent. Aside from the six Sector receptions and all the Big Twelve receptions, you've got a hundred and twenty-odd Divisions, and each of _them_ is throwing a reception. Every theater, auditorium, museum, hockey rink, and barn has already been booked."
"I may have a solution," Gloria told her. "Pug Ellison's family owns that neo-Gothic monstrosity up on the cliff. The Ellisons are here tonight, and Pug's going to introduce me to them. Maybe I can arrange something."
"Gloria, darling, that would be marvelous. I'm off, then."
"Althea?"
"Yes?"
"You were kidding about that utility room, weren't you?"
Althea's violet eyes widened in innocent surprise. "No," she said, "I wasn't."
JILL CLYMER CAUGHT SIGHT OF ALTHEA DANTE emerging from an unmarked doorway on one side of the hall, a languid smile on her face. Jill laughed to herself, amused by her OSI colleague's dedication to the cause. She saw no need for such extreme measures, but didn't mind putting on a bit of a show for the sake of OSI. Considering the public image Gloria had imparted to the OSI, she could hardly have done less.
She could hardly have worn less, at any rate. Her microns-thick smart-fabric dress, featuring random starbursts of silver and gold flashing against an inky backdrop, was cut low to reveal impressive cleavage, both fore and aft. She had never been shy about her physical endowments, but since joining OSI she had come to feel downright dowdy next to Gloria and Althea and, lately, even Petra. Jill had always considered herself to be in the Lion category, but she had to admit that there were advantages to Tiger stripes.
Schmoozing, in any case, was an art she had been practicing since she was six. Her father had been in Parliament for a while, and Jill had grown up at gatherings like this one, where the most important and self-important people in the Quadrant stood around with drinks in their hands and sneers on their faces as they sized up the competition. Jill chatted them up with skill and confidence and made, she felt, a good case for OSI.
She was glad she had accepted Gloria's offer to join OSI permanently following the Sylvania mission. She liked and respected Gloria and believed that OSI could play an important role in Dexta and the Empire. Like her father before her, she believed in good, clean, and efficient government; Dean Clymer had been hounded from office and into an early grave because of that belief, and Jill had entered Dexta in an attempt to carry on the good fight.
She had just finished dictating some notes to her pin-pad when she was approached by a tall, strikingly handsome man decked out in his "Imperials"—black knee boots, tight white trousers, and deep blue tunic, the standard garb for men at formal affairs like this one. Jill had never met him before, but recognized him immediately—Eli Opatnu, the Sector 19 Administrator. He was, she knew, that rarest of animals in the Dexta menagerie, a male Tiger. Rumor had it that he possessed genetic enhancements that allowed him to pump out pheromones on demand, although his confident good looks, trim body, and tighter-than-necessary trousers all suggested that he didn't really need them. He smiled at her, and Jill had to catch her breath.
"Ms. Clymer, I presume?" he asked, extending his hand. Jill nodded and shook it.
"Good evening, Mr. Opatnu," she said. "A pleasure to meet you."
"Likewise. I hear very good things about you, Ms. Clymer."
"It's Jill, and I've heard good things about you, too. And I know you've met with Gloria about the double-flagging problem. That's my project-of-the-moment."
"Yes, I know. Can you tell me if you've made any progress? Naturally, I'm curious, although Gloria has been somewhat reticent about sharing all of OSI's secrets."
"We have a Financial team on Staghorn," Jill told him, "but they haven't really reported much yet. When they do, we'll be sure to keep you informed."
"I'd very much appreciate that, Jill." Opatnu grinned and gleamed, his green eyes all but boring holes into hers. She held his gaze and felt a hot flush creeping into her cheeks and an urgent ache in her stiffening nipples. _Spirit, the man is fast!_ Jill tried to get a grip on herself.
"As you know," she said with what she hoped sounded like professional detachment, "the investigation is centering on Wendover Freight and Storage, which is based in Sector 19. But they have offices here, as well, so as long as I'm here anyway, I plan to pay them a visit."
"Mind if I tag along?"
"If you'd like. Say, the day after tomorrow?"
"Splendid. And in the meantime, perhaps I can lure you away from this den of iniquity and show you some of Central's more intimate and intriguing nightspots?"
Jill shook her head. "Sorry, I'm on duty at the moment. I have hundreds of hands to shake and asses to kiss tonight, I'm afraid."
"As long as mine is one of them," he said, grinning again with full intensity, "I'd be happy to wait until you've fulfilled your duty to Gloria. Perhaps we can even induce her to join us."
_A twofer,_ Jill wondered. Or was she getting ahead of herself? Was she simply reacting to the man's reputation, the way men reacted to Gloria's, or was she responding to Opatnu himself, pheromones and all?
"At the moment, however," Opatnu said, "I need to kiss a few backsides, myself. How about if I look you up later this evening?" He took Jill's left hand, raised it to his mouth, and kissed it gently.
"I'll be here," Jill said noncommittally. Opatnu released his hold on her, smiled again, then turned and strolled off into the glittering throng.
"Steady, girl," Jill told herself.
PETRA WONDERED WHERE THE HELL PUG WAS. He was supposed to be shepherding his parents around the reception, but she had seen him from a distance, and that wasn't his mother he'd had his arm around. It was Steffany Fairchild.
Old friends. Entirely innocent.
You bet.
Her feet hurt in the high gravity and high heels, and she'd had more champagne than was consistent with ladylike deportment. Perhaps that accounted for her growing desire to plant one of those spiked heels in the shapely derriere of Ms. Fairchild, the Spirit's gift to the overbred men of New Cambridge.
But, she reminded herself, she had a pretty shapely derriere of her own, as men had been telling her, one way or another, all evening. And her tits weren't bad, either—small but enticing and entirely uncovered in the low-slung, gold-leafed black pareu Pug had bought for her on a shopping spree the day before. He liked to see her undressed this way, as if she were Gloria's alter ego instead of merely her assistant, and Petra was willing to play along. She had no intention of losing Pug to that bouncy blond airhead or any other rich bitch from his old stomping ground.
A passing waiter paused near her, and Petra took the opportunity to trade her empty glass for a full one. Half-drunk and about 97 percent naked, she was feeling combative and defiant tonight. How had Gloria put it that time, when she tried to explain what it meant to be a Dexta Tiger? Tits, tush, and twat fully mobilized for the sake of Dexta? Fine—and if not for Dexta, then for her rich and handsome boyfriend. And also, perhaps, for herself, the glamorous and sexy new Tiger on the block, no longer the mousy and unsure Lap Dog. And maybe, just maybe, it was also a way of saying, "Fuck you, New Cambridge!"
She wasn't sure, but she thought she might actually hate this goddamn planet. Snobby, disapproving Ellisons by the score; interfering uncles; wayward offspring of the rich and powerful; undead girlfriends; hostile and brain-dead bureaucrats...So far, the only person on the whole planet she had actually liked had been lecherous old Jamie Quincannon. On the fifth floor, with the elevator broken.
And maybe Whitney Bartholemew, Junior, too, who, it had turned out, was not only the grandson of Norman Mingus but also the heir of the man who had run B & Q Shipping fifty-five years ago, and the man who ran it today. She had liked his insolent air and what-the-hell attitude and the hint of contempt for the High Society of New Cambridge. He didn't seem to be here tonight. She needed to talk to him, so maybe she could get Pug to arrange a meeting later in the week. Assuming Pug could be persuaded to neglect his family and friends for an hour or two.
Petra noticed a distinguished-looking silver-haired man staring at her bare breasts. One more Dexta functionary to charm. One more anonymous cog in the vast wheels of Empire. She maneuvered toward him and plastered yet another grin on her face. "Hi," she said brightly, "I'm Petra Nash from OSI."
"How nice for you," said the distinguished-looking man. "I'm Edwin Ogburn, and I'm the President of the Republic of New Cambridge. How are you enjoying our planet, Ms. Nash?"
Not anonymous, then. Not even Dexta. Still, he was just the man she wanted to see. She reached out with her free hand and clutched his upper arm.
"Mr. President," she said, "I wanna file a complaint! It's about your damned gravity..."
GLORIA HID BEHIND SOME OVERSIZED POTTED plants as she spoke into the pin-pad that was part of her jewelry. "Jerome Devers-MacDowell, hyphenated, I think, Level Nine, Assistant Quadrant Coordinator, based on Alhambra Four in Sector 22, has been in Dexta forty-two years, married, wife Ethel, two children, five grandchildren and he has images of them. Nice, unassuming guy, seemed a little starstruck. Send him a signed picture, the usual. Seemed sympathetic to OSI. Myra Chow, Level Eleven, Assistant Sector Supe, Sector 20, Norska Two, divorced, has thirteen-year-old daughter Joanne, who is a big fan, send her the glam package. Myra says she has arguments with her coworkers about me, and about OSI. I seem to be a subject of much watercooler controversy. Myra sees me as an empowering role model, whatever the hell that is. Anyway, she's on my side and should be cultivated."
How many did that make? She had lost track. And here came another one...
This one walked right up to her, stuck out his hand, and introduced himself. He was probably in his fifties, medium height, with a sharp nose, slicked-back black hair, and an unidentifiable accent, probably from somewhere off in Sector 3 or 4, where they spoke an increasingly strange brand of Empire English, when they spoke it at all. "I'm Anton Grosz," he told her. "I'm the Assistant Admin of the ETR."
ETR—the Office of Exo-Technology Review—was one of those small, specialized ecological niches that proliferated within Dexta, whose purview included bits of bureaucratic flotsam that didn't fall under the aegis of the Quadrants. The Empire was expanding in every direction, and regularly absorbed alien civilizations—nearly three hundred of them so far. Such acquisitions inevitably brought with them an array of alien technologies. The Empire needed some way of organizing and controlling the influx of exo-technology, and that job belonged to ETR.
The ETR identified, cataloged, and classified the technological achievements (as opposed to purely scientific ones, which were the responsibility of another Office) of the new civilizations that had joined the Empire, willingly or otherwise. The task sounded larger and more interesting than it really was. Fully 94 percent of all alien technologies proved to be nothing but idiosyncratic elaborations of existing Terran processes or devices. Technological evolution followed much the same pattern among different civilizations, just as biological evolution tended to reproduce optimal design features for similar environments on different worlds. Fish, for example, looked pretty much the same on most planets; so did can openers. So the ETR identified, cataloged, and classified a lot of can openers.
The Big Twelve corporates looked over the ETR's shoulder as it checked the new technologies to see if they overlapped with existing Terran patents. If they did, they were denied Export Permits, effectively restricting the technology to its homeworld. If something genuinely new or useful turned up, the Big Twelve were free to strike whatever deals they could with the new worlds. Soon, the new technologies would become available throughout the Empire, to the considerable profit of the corporates who landed the contracts. Meanwhile, new markets opened up for the export of Terran technologies.
ETR also tested the new technologies for safety, environmental compliance, and what was known in the trade as "weirdness." Perhaps once or twice a century, some truly weird and unexpected alien technology would turn up and suddenly revolutionize the way Terrans did things. The mass-repulsion devices that made skimmers and null-rooms possible had arrived that way, the fruit of a victory over a race of intelligent insects in the twenty-fourth century, back in pre-Imperial days.
Gloria had already targeted ETR as a potential ally, so she was pleased to meet Grosz. But she was more than a little surprised when Grosz said, "Ed Smith says hello."
"Ed Smith? You mean...?"
Grosz nodded solemnly. "I'm your contact inside Dexta," he said. "Obviously, you can't go on having public meetings with Mr. Smith, although he hopes to meet with you privately after your return to Earth. When you have a message for him, or he has one for you, I'll be the go-between."
"I see," said Gloria. She stared at Grosz and tried to imagine him as a daring undercover agent, but couldn't quite manage it. "So, the zamitat has its hooks in ETR, huh?"
"We have an understanding," Grosz said modestly.
Gloria was well aware that the zamitat had people inside Dexta, but it felt odd to be meeting one of them. This, after all, was precisely the sort of thing the OSI was supposed to be in business to correct.
"The Big Twelve have a formal presence," Grosz went on, "but the zamitat has its own place at the table. Every so often we find a piece of exo-tech that could be of interest to them."
"And they pay you to tip them off?"
"They pay me to represent them at ETR. If you disapprove, you should have thought of that before now."
"No, no," Gloria said, anxious not to offend Grosz. "It's not that I disapprove, exactly. It's just that I'm a little new to this sort of thing."
"It will grow on you," Grosz assured her. "Anyway, I have a message from Mr. Smith. He says that there was definite zamitat involvement in the historical matter you inquired about. He hasn't been able to ascertain any details yet, but he's pursuing the matter and will keep you apprised."
"Thank him for me. And thank you, too, Mr. Grosz."
"You're very welcome, Ms. VanDeen. And now, I shall slink off into the night, intent upon my dark errands." Grosz gave her a modest bow, then darted away to the other side of the potted plants.
Well, well. The zamies had something to do with the Savoy shipment. Details to follow. Based on what Petra had told her, it was beginning to seem likely that the zamitat had either diverted or in some manner controlled that final B & Q shipment to Savoy. If that were the case, then perhaps Ed Smith or his minions could track down the shipment and provide some indication of how those weapons had wound up in the hands of PAIN. Petra had yet to go through all the data she had downloaded from the B & Q files, so perhaps additional information would turn up from that source.
Gloria felt a sense of relief at their measured progress. There had been no new reports of terrorist attacks, and no one else had taken a shot at her, but the thought of all those weapons in the hands of enraged anarchists continued to be troubling. The Quadrant Meeting was an obvious target, and despite the heightened security on New Cambridge, a determined PAIN attack might do serious damage—not only to Dexta, but to the reputation of the Empire itself.
But it was the reputation of OSI that was her prime concern tonight. Althea, Elaine, Jill, Petra, and the two guys were busily spreading the word, and Gloria was grateful for their presence. They took some of the pressure off her and reinforced OSI's image as the sexiest and most appealing branch of Dexta. Just now, she caught sight of a very nearly naked Petra chatting with a group of three men. And over in another corner, Elaine Murakami, in her hardly there black minidress, seemed to have captured the complete attention of three more men. Women generally seemed to be responding favorably, too, if Gloria's own contacts so far were a reliable indicator; Brent and Darren seemed to be making quite an impression. OSI was something new under the Dexta sun, and despite the opposition of the Quad Admins and other entrenched powers, it was clearly attracting a lot of interest, if not active support.
Gloria took a deep breath, fixed a smile on her face, and stepped out from behind the potted plants. Back to work.
PETRA'S PAREU HAD FINALLY FALLEN OFF AND she had simply draped it over her shoulders like a shawl, leaving the rest of her body uncovered. Just like being back at Elba's, she thought, or at the Ellisons' party. The center of attention again—or _a_ center of attention, anyway. Gloria was here, of course, so no other woman really had a chance. Except maybe the lovely Steffany Fairchild.
She spied Pug, along with his parents and Uncle Benedict, his arm around Steffany, off in a corner with Gloria, Althea, Elaine, Jill, and Eli Opatnu. It was late, and the party was emptying out, so Petra was able to chart a fairly direct course to them. She arrived at the corner, swaying slightly on her high heels and aching feet. Pug grasped her arm and pulled her away from his family; his father looked pleased to see her but his mother wore a disapproving frown.
"Are you drunk?" Pug demanded in a harsh whisper.
"What if I am? And what is _she_ doing here?"
"She came with her father."
"Then where is _he_?"
Before Pug could answer, Steffany joined them. She cast her eyes over Petra's body and offered a supercilious smile. "Good evening, Petra," she said. "Still showing off your dowry, I see."
"Dowry?" Petra replied blearily. "What? Is that some kind of crack? Oh, wait, I get it. Dowry. Cute. Is that what you call it? Perfect. I bet if I pulled on your arm, hundred-crown coins would drop out of _yours._ "
_"Really,"_ Steffany sniffed, and turned away.
"Come on, I'm taking you home," Pug told her.
"No, you're not," Petra responded. She slipped away from him and sidled up next to Jill Clymer. "I'm staying here with my friends!"
"Uh...actually, Petra," Jill said, "Eli was just going to take us all out to a club."
"Great. I need more champagne."
"Don't you think you've had enough?"
"Not yet," Petra declared, glaring at Pug.
JILL ASSIGNED HERSELF TO TAKE CHARGE OF Petra as they made the rounds of Central's nightspots. Opatnu was their knowing guide, evidently familiar with all the more intriguing hideaways in the city. He somehow managed to pay equal attention to all five women, and Jill found herself wishing that the other four were somewhere else. When she danced with him, his charm, grace, and outlandishly good looks all but melted her resistance, and she snuggled up close to him while he rhythmically stroked her nearly bare bottom. But all too soon, the dance ended, and Jill was back at their table keeping Petra company while Eli whirled around the dance floor with Elaine Murakami, who had doffed her minidress and was as naked as Petra.
Then it was on to another club, and another, this one darker and more intimate than the others. Opatnu excused himself for a few moments, leaving the women from OSI to themselves. Gloria was smiling happily. "Thanks, ladies," she said. "You did a great job tonight."
"It was all due to your inspiring leadership, Gloria, dear," Althea replied. "And getting the Ellisons to agree to host our reception was nothing less than a coup."
Petra looked up at the mention of the Ellisons. "Yeah," she said, "you'll love it there, Althea. Did I tell you, Pug has a bed the size of Weehawken? O' course, if you wanna use it, you'll probably have to kick Steffany Fairchild out of it." Petra took another gulp of champagne, then nearly toppled out of her chair. Jill caught her by the arm and propped her back up.
Opatnu returned, took a seat, and grinned like a cat who had swallowed a medium-sized canary. "Got something special," he said, and held out his hand. Resting in his palm were six purple lozenges.
"Twenty-nine!" Elaine cried.
Opatnu shook his head. "Better," he said. "It's Forty-eight."
Gloria's eyes widened. "Forty-eight? How did you...?"
He smiled modestly. "I know some people," he explained. "Works the same way as Twenty-nine. Just make sure that you don't crunch down on it. And I've booked a null-room, for those who are interested."
Opatnu extended his hand. Gloria, Elaine, and Althea didn't hesitate. Petra looked at the lozenges briefly, but didn't show much interest, so Opatnu moved on to Jill. She stared at his offering for a long moment. She'd tried Twenty-nine a few times, and was no stranger to null-rooms. But something about it all did not feel quite right to her. Perhaps she just didn't want to have to share Opatnu.
"I think I'll just stay here and keep an eye on Petra," Jill said at last.
"Are you sure?" Opatnu asked. He looked at her with an expression of profound disappointment on his handsome features.
"You go ahead. We'll be fine."
"Some other time, then," Opatnu said.
"We'll see," Jill replied.
Opatnu nodded, then got to his feet. Gloria, Elaine, and Althea followed him across the floor to the null-room. Jill watched them go with decidedly mixed emotions.
Petra looked up, looked around, and asked, "Where'd everybody go?" Before Jill could answer, Petra quietly passed out, slumping forward onto the table.
Jill appropriated Petra's champagne glass and offered a toast. "Here's to OSI," she said. "The Fifth Quadrant." Softly, she added, "Spirit save us."
PETRA AWOKE THE NEXT MORNING TO THE sound of her beeping wristcom. "Yeah?" she mumbled.
"Petra Nash?"
"I think so."
"Good morning, Petra Nash. This is Whit Bartholemew. How are you today?"
That was a question Petra was not prepared to answer just yet. She felt too awful to be able to calibrate precisely how awful. The only response she could manage was a weak groan.
"Sorry I missed you at the reception last night, but I was called away on business," Bartholemew said. "I was truly looking forward to seeing you. I hear you were the toast of the Quadrant."
"I was?"
"So they say. But I would prefer not to rely on mere gossip. I'd much rather see for myself. Would you consider having lunch with me today?"
"Uh..."
"Wonderful. I'll send a limo for you, say, about noon?"
"Uh..." Petra looked around for the first time and realized that she was not at the Ellisons'. She seemed to be in a bed in Gloria's suite at the hotel. "I'm at the Imperial Cantabragian," she said.
"Yes, I know. Noon, then?"
"Uh...yes, noon is fine."
"See you then, Petra Nash."
Petra tried to get up, didn't quite make it, but managed to swing her legs around and sit on the edge of the bed while the room gradually stopped whirling around her. She took a few deep breaths and noticed her pad on the bedside table, glowing orange to announce a message. She reached out and keyed it.
"First things first," Gloria said brightly. "There are some No-Regret tabs in the bathroom. Elaine and I are off to the committee venues. I got a call from Pug, who says he hopes you're feeling better and that he'll be checking out Stavros & Sons today. He says he'll see you tonight at the Ellisons'. I don't know what you had planned for today, but if you need to take the day off, go ahead. Oh, and Elaine says you're welcome to borrow some of her clothes. Have a great day, and I'll talk to you later."
Petra made her unsteady way into the bathroom, found the pills, and swallowed two of them. Then she stepped into the shower, hoping she would be lucky enough to drown. She didn't, and by the time she turned the shower off, the pills had taken effect and she was feeling marginally human, if not exactly chipper. She remembered most of what had happened last night and gave the bottle of pills a skeptical look. "No regrets, huh?"
She supposed that she had not completely disgraced herself, but she had probably come pretty close. Prancing around in the nude in front of all the Dexta bigwigs of Quadrant 4 might not have been a prudent career move, now that she considered it. On the other hand, with Gloria there, it was possible that no one had even noticed. Gloria was like a black hole, sucking in light and energy from the minor bodies that orbited her, and sometimes it was impossible for Petra not to resent her. Still, Gloria couldn't help being what she was, any more than Steffany Fairchild could help being a tall, blond, big-titted bitch. Petra, fated to be cute and little, felt that she had to resort to extreme measures in the presence of such overpowering competition.
If Pug thought he could take her for granted and cuddle up to Steffany while Petra labored away for Dexta and OSI, maybe he needed a little reminder that Petra Nash was nobody's doormat. Whit Bartholemew seemed to think highly of her, at least, even if nobody else on this fucking planet did. He'd missed the show she put on last night? Okay, she would put on the same show for him today. She didn't bother with borrowing something from Elaine, but found her black-and-gold pareu and knotted it recklessly low on her hips. Just the thing for a casual lunch.
Petra ordered coffee from room service, and while waiting for it to arrive, she rummaged around in Gloria's room until she found the inevitable package of jigli. She lit up one of the potent cigarettes and settled comfortably in a chair to smoke and stew. By the time the coffee arrived, she was tingling all over with sexual arousal and righteous indignation.
THE LIMO DRIVER EXPLAINED TO PETRA THAT she would be having lunch in Bartholemew's office rather than a restaurant. That was probably just as well, Petra reflected, as she got out of the limo in front of the Bartholemew Building in downtown Central. The provincials of this two-crown world didn't seem to be used to the sight of sophisticated Manhattan ladies striding past them at high noon, bare-breasted and practically nude. Come to think of it, this would have been pretty extreme even for Gloria. So be it; Petra Nash, Dexta Tiger, didn't care.
She arrived at the top floor (the elevators in this building, thank the Spirit, were working) and was quickly ushered into Bartholemew's private office by a secretary who kept giving her disapproving sidelong glances. Whit Bartholemew greeted her with a wide smile and wider eyes.
"I didn't have a chance to change after last night," Petra explained as Bartholemew kissed her extended hand.
"Why change perfection?" Bartholemew said as he focused his gaze on her pert breasts and sensuously bared belly. Petra felt a surge of jigli-fired tingles.
Bartholemew took her hand and led her on a short tour of his palatial, fiftieth-floor office as he explained that Bartholemew Enterprises, of which he was president, was really a holding company that managed fourteen different concerns. They ranged from the moribund but occasionally useful B & Q Shipping to a financial company that underwrote construction projects throughout the Quadrant. "My father, of course, is the one who built this overweening enterprise," he said. "And through no fault or merit of my own, I now sit in his big chair and crack the various whips that keep the serfs busy and productive."
"You sound as if you regret it."
"Endlessly. I tell you, Petra Nash, if not for the disadvantages of my birth, I might have amounted to something in life." He led her to a small table, where wine and a salad course were waiting for them.
"Well," Petra said as she sat down, "it's not too late. I mean, you could still just chuck it all and go...do what? What would you have done with your life if you hadn't been burdened with wealth and position?"
"Interesting question, that," Bartholemew said as he poured the wine. "Poet? Politician? Proctologist? In a way, I've had to be all of those things anyway. Especially the last one. You wouldn't believe how many assholes I have to deal with."
Petra smiled. "Me too," she said.
"I shouldn't doubt it. Tell me, whatever possessed you to join that gang of fascist file clerks?"
"Oh, it's not that bad. Dexta's just another bureaucracy, with all the faults of any bureaucracy. I joined because I wanted to make something of myself and help run the Empire. Maybe do a little good in the galaxy. Why are you and your mother so down on Dexta, anyway?"
Bartholemew sipped a little wine and leaned back in his chair. "You've met my grandfather?"
Petra shrugged. "Not really. But Gloria knows him well and thinks he's a wonderful man."
"Gloria, no doubt, would. Actually, I barely know the man. I've only met him three times in my life, and the last time was over twenty years ago. But I know what he is, what he's done. He's a tyrant, nothing less. He absolutely controls our pathetic self-indulgent, milksop Emperor, and has the goods on everyone in Parliament. I tell you, Petra Nash, _no one_ can be trusted with power, let alone _absolute_ power. Least of all, Norman Mingus."
"His power is hardly absolute," Petra pointed out. "He has plenty of trouble just controlling Dexta, from what I hear. As for the Emperor and Parliament...well, we do have an _Imperial Code,_ don't we? Dexta is just one part of the balance of powers."
"Right out of the textbook. Is that what they taught you back at dear old Alexander Hamilton High School in lovely Weehawken? Odd that they should name the place for a man who was killed there, don't you think? Why not Aaron Burr High School?"
"You've been checking up on me," Petra said with some annoyance.
"From the moment I met you," Bartholemew agreed. "You intrigue me, Petra Nash. You are intelligent, witty, charming, and beautiful in a way that your famous boss could never be. You're a real person. I like real people. I've met so few of them in this charade I call my life."
"Why are you so bitter?"
"Better to ask why _everyone_ isn't that bitter," Bartholemew said. He drank some more wine. "Everyone certainly has cause to be in this benighted Empire of ours. Why aren't _you_ bitter, Petra Nash? You certainly have cause to be, after all the miserable cards you've been dealt. Absentee father, shrew of a mother, egotistical, self-absorbed boss, weak-kneed, two-timing boyfriend..."
"Now, wait a minute!" Petra felt a hot flush of anger burning her cheeks. "Ever since I got here, you've been going out of your way to insult me, my job, and all the important people in my life. What the hell gives you the right to set yourself up as the Universal Judge?"
"Nothing," Bartholemew admitted. "You're quite right. I'm no more qualified for that job than anyone else. And that's my whole point. My contempt, I assure you, is not directed at you, or Gloria VanDeen, or Pug, or even Norman Mingus. It's a purely impartial contempt, aimed at one and all—Whitney Bartholemew, Junior, not excepted. Forgive me, I didn't mean to upset you."
"I'm not upset."
"Yes, you are. There's some color in your cheeks. And your exquisite nipples are so stiff, they look like gun barrels aimed straight at my heart." Bartholemew clasped his hands over his chest in mock agony.
Petra laughed, in spite of herself. She took a sip of wine and stared at the man across the table from her. She didn't think she'd ever met anyone quite like him. If he was obnoxious and hurtful, he was also brutally honest. There was something to be said for that. Bartholemew seemed to be a man who was determined to live without illusions—or maybe he just preferred the illusions he created for himself.
Bartholemew hit a button on his wristcom, and an instant later a wheeled robot dumbwaiter came in and served the main course of fish, rice, and vegetables. The robot departed, and Bartholemew poured more wine.
"As long as I've already provoked your pique," he said, "allow me to annoy you even more. I must confess that I didn't invite you here simply so I could insult you and stare at your lovely face and that splendid little body—which, by the way, I am extravagantly grateful to you for revealing so completely. No, I have another motive, entirely. I understand that you paid a little visit to that old fart Quincannon."
"Yes. I needed some information about B & Q Shipping for an investigation I'm doing. I didn't know that you owned the company."
"But now you do know it. And much more. You downloaded all of the company's files, sixty years of them."
Petra nodded. "I needed to find out about a B & Q shipment in 3163. I didn't want to spend hours looking for exactly what I needed, so I just downloaded everything."
"So I gather. That nitwit Quincannon apparently didn't realize that something like that would immediately show up in our master computer. I would never have approved letting you do that, had I known."
"It wouldn't have mattered whether you approved it or not," Petra told him. "I'm investigating a possible link to terrorist activity. I didn't have a warrant, but I could have gotten one easily enough."
"Possibly. That's neither here nor there. If you are investigating something that happened in 3163, you hardly need _all_ of B & Q's records, do you?" Bartholemew offered her a friendly smile.
"I told you, I didn't have the time to do a specific search. But once I've downloaded everything from my pad to the office computer, I'll be able to find the stuff I need."
"Fine. And what of the stuff you _don't_ need?"
Petra shook her head. "I have no interest in that."
"In that case, before you dump all of that information into the maw of Dexta's computers, would you be kind enough to separate the wheat from the chaff, then permanently dispose of the chaff?"
"Why? I told you—"
"Yes, I know, you have no interest in that. But someone else might. That's my problem. You see, some of the material that is of no interest to you contains information that could be, shall we say, _embarrassing,_ to B & Q Shipping and Bartholemew Enterprises and to me, personally."
"Look, if you have some skeletons in your corporate closet, don't worry about it. I couldn't care less. The only thing I'm concerned with is that shipment to Savoy in 3163, and Pug and I are the only ones who are going to be using that data. Your secrets, whatever they may be, are perfectly safe."
"I wish I could be certain of that. But the ugly truth is that it's not simply a matter of a few skeletons in a closet. Apparently no one has told you about my father."
Petra frowned. "What about him?"
Bartholemew drank some more wine, then sighed. "You see," he said, "the fact is that my sainted father, Whitney Bartholemew, Senior, spent his entire adult life working in and for the zamitat. Those records, Petra Nash, do not constitute a few skeletons in a closet. They constitute an entire fucking boneyard."
"Oh," said Petra.
"Indeed. So you can understand my reluctance to let you drop the whole sordid mess into the hopper. It's not simply that _I_ would be embarrassed. I'm not even in their damned cartel. But if the zamies ever got wind of it, there could be some unpleasant consequences for everyone concerned. Me, Quincannon...possibly even you and Pug. You unwittingly downloaded a time bomb into your little pad. For my sake, and for your own, it would be wise of you to get rid of all that information as quickly and quietly as possible."
"I see," Petra said.
"You'll do it, then?"
"I'll have to think about it. What you're asking is, in itself, illegal. I mean, just _asking_ me to do that is a felony. Actually doing it...Spirit, I just don't know." And she didn't. Petra had never been confronted with anything like this and didn't have a clue how to deal with it. The zamitat? What had she gotten herself into? She felt a sudden chill up and down her bare back.
They finished their lunch in relative silence. Bartholemew opened another bottle of wine, poured some more, then led Petra to a corner of the vast office that appeared to be a small library. He put his arm around her back and gently massaged her right shoulder.
"You needn't be afraid of me," he said.
"I'm not. But I am curious. Your mother married a man in the zamitat. Mingus must have known. Is that what went wrong between them? Did he object to her marriage?"
To her surprise, Bartholemew erupted in a genuine laugh. "Object? Hardly! He arranged it. Or more precisely, he forced her into it. More precisely still, he sold her to that fucking thug I am pleased to call my father."
Petra looked up into his eyes. "But why?" she asked him.
Bartholemew shrugged. "It suited his needs. My grandpa needed something from my daddy, so he gave him my mommy, and everyone lived happily ever after. Heartwarming, isn't it?"
"I don't understand. Why would she go along with a thing like that?"
"I can see that you don't understand power. I mean serious, enduring, political, economic, and societal power. Stay with Pug a while, and maybe you will."
Petra thought about it for a moment and nodded. "I think I see what you mean," she said. "I've seen the kind of power the Ellisons have. I mean, I don't think Pug really wants to take that job on Pelham, but..."
"But he will," Bartholomew said with flat confidence. "Just as my mother married old Bart. She was in love with another man at the time—and that, too, had been arranged by my kindly old grandfather. But then his needs changed, so he redirected his daughter's life. If he hadn't, I wouldn't have been Whitney Bartholemew, Junior. I'd have been Cornell DuBray, Junior."
"DuBray?"
"None other. He was Mingus's assistant at the time, and it must have seemed an obvious and convenient match. But then, events intervened. Mingus made DuBray break off the engagement, then delivered his innocent young daughter into the waiting and eager arms of the biggest bastard in the Quardrant. And Mother went along with it, because what choice did she have? None, really; no more than those ancient princesses who were bartered off to unholy wedlock with foreign kings and potentates because it suited someone's needs. The needs of power."
"I can understand the marriage, I guess," Petra said. "But why did she stay with him all those years?"
"You don't divorce a man like Whitney Bartholemew, Senior. And, in time, a bond developed between them. Call it love, if you like. My mother was beautiful, glamorous, and well connected; my father was handsome, powerful, and domineering. She made him look respectable, and he made her feel needed. But she never forgave her father for what he did to her."
Petra looked up at Bartholemew. "It must have been difficult for you," she said.
"Compared with what?" Bartholemew shrugged. "It was the only life I knew. I suspect that it was actually pretty easy compared with, say, growing up in poverty in a broken home in Weehawken, New Jersey."
"Oh, that wasn't so bad, either. Not really. It was the only life _I_ knew."
"And yet now, you seem to be prepared to embark on an entirely different sort of life. Consort, or perhaps wife, to the scion of one of the wealthiest and most powerful families in the Empire. Are you sure that's what you want, Petra Nash?" Bartholemew raised his right hand and stroked her gently under her chin with his index finger. She looked up into his dark eyes.
"I...I'm not sure," she said. "I love Pug, but..."
"And I'm reasonably sure that Pug loves you, as well, as far as he is able. And yet, at this very moment, he's with Steffany Fairchild."
Petra pulled away from him. "No, he's not," she insisted. "He's busy with Dexta work."
"He was, this morning. Now, he's with Steffany."
"I don't believe you. How could you know that?"
"Because he told me. I called him this morning when I was trying to find your commcode. If you don't believe me, why don't you call him yourself?"
Petra almost did, but thought better of it.
"Pug is with Steffany," Bartholemew said, "and you are here, with me. Don't try to tell me, Petra Nash, that you came here dressed like that just to have a social luncheon."
Bartholemew gently ran his hand over Petra's bare breasts, toying with her nipples. Petra's breath came in short, tense gasps as her jigli-enhanced nerves tingled in spasms of intense pleasure. His hand moved downward, probed her belly button, then slid on across her belly and down to where her pubic hairs curled over the top of the knot in her pareu. He paused there for a moment, then deftly unknotted the pareu and cupped her groin with his palm.
Petra tilted her head back as he stroked her and finally breathed, "You're right. I didn't." She reached for him, pulled him down to her, and kissed him.
GLORIA'S FEET HURT. SHE DIDN'T LIKE HIGH-GRAVITY worlds; humans weren't designed for them. But humans were nothing if not adaptable, and they lived on Empire planets with gravitational forces ranging from .1 G to 1.7 G. In time, evolution would sort things out, but that thought was no comfort to her as she roamed the corridors of the Convention Center on her high heels and aching feet. She had sat in, briefly, on a few of the many committee meetings, but most of the day had been spent schmoozing in the corridors, selling the delegates on the benefits and potential of the OSI.
A gaggle of media reps had trailed her, off and on, but she mainly ignored them and concentrated on buttonholing Dexta people and chatting them up. She wore a sheer-to-the-point-of-invisibility white blouse with a wide, deep neckline, a matching skirt slit to the hipbone, and a businesslike dark blue blazer. Everyone could see everything she had, but she still managed to look brisk and efficient. It was a combination she knew worked well for her and somehow symbolized the image she was trying to promote for OSI. We're open and friendly, we have nothing to hide, and we get the job done.
Last night, she knew, had been a triumph. Jill, Althea, Elaine, Petra, and the guys had put a personal, smiling face on OSI for hundreds of Dexta delegates who would otherwise have known it only from rumor and reputation. Oh, Althea may have overdone it a bit, as usual, and Petra had been too drunk to function by the end of the evening, but overall, they had accomplished everything she had hoped for. And she hadn't done badly herself.
And afterward...Eli Opatnu and Forty-eight! The new drug was everything it was said to be, and more. Gloria closed her eyes for a second and shivered at the memory of the fantastic sensations that had coursed through her for what seemed like hours. Spirit, that was powerful stuff! And Eli was pretty powerful stuff, too; he had more than satisfied Gloria, Elaine, and Althea in the null-room. It was a shame that Jill had declined the opportunity and Petra was too far gone to participate. They might have learned something from the experience.
Eli, Gloria realized, was her opposite number, her mirror image. He was concentrated sexual energy in a package of stunning beauty. Perhaps her reaction to him was the female equivalent of what men felt with her. She felt no desire to form a lasting relationship with him; that, she was certain, would destroy them both. Like Forty-eight, Eli Opatnu was a transcendental treat, but not something you'd want to take with breakfast every morning. But there was no reason they couldn't continue to see each other from time to time and delight in each other's gifts.
"Ms. VanDeen?" Gloria was roused from her reveries by a trim-looking, dark-haired woman standing in front of her.
"Yes?" she said, smiling pleasantly.
"I'm Harriet Graves, Level Eleven, Sector 24 Social Services Coordinator."
"Pleased to meet you, Harriet."
"Ms. VanDeen, I just wanted to tell you that I think you're a disgrace!"
Gloria raised an eyebrow. "Oh?"
"You've cheapened and degraded yourself and Dexta. You've harmed every woman in the organization with your narcissistic display and self-indulgent theatrics. As if women had nothing to offer but sex!"
"Well," said Gloria, "you're entitled to your opinion. But I think the record shows that I've offered a lot more than just sex. I'm sorry if my personal style offends you, but, you know, the OSI has a great deal to offer Dexta. We've been able to solve problems that the traditional Dexta structure was incapable of dealing with, and as the Office expands, we'll provide new opportunities for creative and flexible resolution of—"
"Oh, don't give me any of your OSI garbage, Ms. VanDeen! I don't care a fig about your Office of Strategic Intervention, and the truth is, neither do you! All you care about is making a spectacle of yourself and getting every man in the Empire to lust after you!"
"And what would be wrong with that?" Gloria asked with a bemused smile. "Do not deny yourself joy, Ms. Graves, it's the wellspring of—"
"And don't give me any of your Spiritist crap! I'm a decent, God-fearing Christian woman! Avatar of Joy, huh? Hmmpf!" Ms. Graves whirled around and stomped away in high dudgeon.
A small crowd had gathered during the confrontation. Gloria turned to the onlookers and said, "Well, you can't please everyone, but I think—"
She was interrupted by a loud, sharp _crack-boom!_ The unmistakable signature of a plasma discharge came from somewhere down the corridor. Gloria turned to look, but before she could see anything, two of Arkady Volkonski's Bugs lifted her off her feet and whisked her away. Volkonski himself was suddenly there, directing them through an unmarked door and into what appeared to be a supply closet crammed with janitorial robots, jugs of cleaning fluid, and an array of mops. Volkonski dashed off, slamming the door behind him.
Before Gloria could say anything to the two Internal Security men sharing the closet with her, the door opened again. Two more Bugs herded their charge into the tiny room. Gloria suddenly found herself face-to-face with Cornell DuBray.
They stared at each other in awkward silence for a few moments, then DuBray turned to his security men and gave a subtle nod. They opened the door and departed. Gloria gave a similar nod to her own Bugs, and they left the room and closed the door behind them.
"I hear you've been busy, VanDeen," DuBray said.
"Just showing the flag for OSI," Gloria replied.
"Showing a lot more than that," DuBray said. "I heard your little exchange with that Graves woman. She's not alone, you know. For every slavering man you win over with your charms, you probably alienate two women."
"Not by my count. Most of the women I've met here seem to think I'm a positive influence. They like to see a woman succeed on her own terms and not have to kowtow to anyone who happens to have a Y chromosome."
"Perhaps. In the end, it won't make any difference. The Quadrant Administrators and most of the Sector Administrators are still adamantly opposed to the whole concept of the OSI."
Gloria nodded. "I'm sure that's true. But the thing is, DuBray, Dexta consists of tens of thousands of men and women who are not Quadrant or Sector Administrators. I know you think you run the organization, and maybe you do in a formal sense. But the reality is that Dexta couldn't function without all those lowly Nines and Twelves and Fourteens that you think don't matter. They _are_ Dexta."
"That's a romantic and unsophisticated view, VanDeen. The fact is, when I give an order, those Nines and Twelves and Fourteens obey it. Even glamorous, self-important Tens will obey it, if they know what's good for them."
Gloria shook her head and smiled grimly. "You know what your problem is, DuBray?" she asked him. "You've been so high for so long that you've forgotten what it's like to breathe air that isn't rarefied. You never come in contact with anyone who isn't intimidated by you. But all those faceless Dexta drones are _people,_ with their own hopes and aspirations and agendas. Their own _lives._ They aren't just names in organizational charts. They respond to forces that have nothing to do with bureaucratic imperatives raining down upon them from the heavens above. They _like_ me, DuBray—and they don't like _you._ "
"Perhaps not," DuBray conceded. "But they fear me."
Gloria crossed her arms and cocked her head at a defiant angle. "I don't," she said.
"Then you're foolish."
"Oh? Why, is Erik Manko out of the hospital?"
"This has nothing to do with Manko. That was just a little warning shot across your bow, VanDeen. You should have taken heed. Much worse things can happen to you."
"Well here's a little warning shot of my own, DuBray. My people have established that you signed off on that missing shipment to Savoy in 3163. And I also know that there was zamitat involvement. If I can establish a connection..."
DuBray's bodily reaction was so sudden and sharp that Gloria was certain that her warning shot had, in fact, scored a direct hit. He stiffened, arched his back, and scowled at her.
"You have no idea what you're dealing with here, VanDeen. None! Take my advice, and—"
Gloria never found out what DuBray's advice would have been because Arkady Volkonski abruptly opened the closet door. "You can come out now," he said. "Excitement's over."
Giving DuBray a frosty parting glance, Gloria stepped out into the corridor. "What happened?" she asked Volkonski.
Volkonski shook his head in disgust. "Two undercover Security men," he said. "Each thought the other looked suspicious. So they shot each other. The surprising thing is that it doesn't happen more often."
"Were they seriously hurt?"
"They'll recover. One of them will need a new arm, the other a new ear. Fortunately, their incompetence extended to their marksmanship."
DuBray and his two Security men brushed past them and started down the hallway. Gloria watched him go, feeling a subdued thrill in her gut. She knew now, as surely as she was standing there, that she was going to bring that son of a bitch down.
PETRA SULLENLY SOAKED IN THE TUB IN HER rooms at the Ellison mansion, staring at the ceiling. A mural here, too: water sprites and mermaids. Did the Ellisons control the aquatic mammal franchise on this planet, too, she wondered?
They seemed to control practically everything else. Mainly, they controlled their son. She was certain now that Pug would take the Pelham job. How could he not? Whit Bartholemew might be a cynic, but he was perceptive. Like Saffron Mingus long before him, and countless others, Palmer Ellison would wind up doing what was expected of him. Enduring dynasties like that of the Ellisons did not permit their substance to be frittered away by independent offspring. Even Whit, for all his contempt and bitterness, had wound up running the family business.
She could see the future as clearly as she could see the water sprites on the ceiling. Pug would become Imperial Secretary under Uncle Benedict, spend a few years on Pelham, then move onward and upward in Dexta until he was, probably, about thirty. He'd have stalled at about an Eleven or Ten by then, or at least, failed to advance to the higher levels as quickly as his family would prefer. They wouldn't let him waste more than a few years in Dexta before they pressured him back into the family orbit. By the time he was thirty-five, Pug would be a senior vice president in the Ellison empire, and eventually, he'd take the reins from his father, just as Whit Bartholemew had.
And she could be right there beside him. If that was what she wanted.
Unless Pug decided that Steffany Fairchild would make a more appropriate companion on such a journey.
Not for the first time, she wished they had never come to this goddamn planet. Ever since Sylvania, it had been just the two of them, and it had been wonderful. Now, Steffany Fairchild and Whit Bartholemew had materialized in the midst of their lives, complicating and confusing everything.
She wasn't sure if she had simply been getting even with Pug or whether she felt something deeper for Bartholemew. He was a fascinating but frightening man. Petra was flattered by his attentions and couldn't deny that she had enjoyed making love with him—more than she ever had with Pug, frankly. Maybe it was just the jigli, or the delicious tang of naughtiness, but there was something about Bartholemew that reached a part of her that Pug had never touched. Nor had any other man.
Petra got out of the bath, stepped in and out of the stato-dryer, then paused in front of a full-length mirror. She spent a long time standing there, staring at her naked reflection, wondering just who that person was.
A rap at the door roused her from her meditations. She threw on a robe, went to the door, and found Standish, one of the Ellisons' omnipresent butlers, waiting there. "Dinner in ten minutes, Miss Petra," he said.
"Would you make my apologies to the Ellisons, please? I'm not feeling well. Do you suppose I could be served here in my room?"
"Certainly, Miss Petra." Standish departed, Petra closed the door, and slumped down on the immense bed. She didn't feel like having to deal with the Ellisons this evening. Let them think she was still hungover. Let them think whatever the hell they wanted.
PUG CAME IN MUCH LATER AND FOUND PETRA staring at vid coverage of the Quadrant Meeting. He watched over her shoulder as their boss gazed into the imagers, smiling and sexy, and gave her standard OSI spiel. Gloria seemed to be getting more than her fair share of vid time, but the coverage eventually moved on to other matters. "Makes you proud to be in OSI, doesn't it?" Petra said without looking up from the screen.
Pug didn't rise to the bait. "I checked out Stavros & Sons today," he said. "No Stavros, no sons. They were absorbed years ago by a division of Trans-Empire. Apparently they purged all the old files. I didn't get anything because there was nothing left to get."
"Took you all day, did it?" Petra turned her head and looked up at him.
Pug looked back at her. "Some of it," he said at last. "And how was your lunch with Whit?"
"We had fish," Petra told him. "What did you and Steffany have?"
"Make you a deal," Pug said. "You don't ask me about Steffany, I won't ask you about Whit."
"Sounds reasonable to me." Petra got to her feet and stared at Pug. "Why didn't you tell me Whit's father was in the zamitat?"
"I didn't see that it was relevant."
"I was investigating B & Q Shipping, and you didn't think it was _relevant_?"
"I didn't realize B & Q was a Bartholemew company. If I had, I would have told you. But speaking of B & Q, have you deleted those files yet?"
Petra opened her mouth, then closed it again without having said anything. She turned sharply and marched over to a dresser, seized her pad, and held it up for Pug to see.
"It's all in here," she told him. "I haven't deleted anything, and I'm not going to."
"Petra, you have to—"
"Don't tell me what I _have_ to do! First Whit, now you! What the hell is this, anyway?"
"Look, Petra, be reasonable. All you need is the files from 3163."
"I don't know _what_ I need because I haven't looked at it yet. Spirit, Pug, how can you ask me to delete information from an official Dexta investigation? Don't you realize what you're asking?"
"What I realize," he said evenly, "is that you inadvertently downloaded information that could be very damaging—and dangerous. Do you have any idea what could happen if the zamies ever found out what you've got?"
"Why would the zamies find out? Whit certainly isn't going to tell them. Or is there something I don't know? Maybe something else you didn't think was relevant? Is _your_ father in the zamitat, too?"
"Of course he's not!" Pug glared at her from across the room. "My family's business is entirely legitimate. Or as legitimate as any business ever is."
"Then why are you so upset about this? What difference does it make to you if I accidentally stumbled across something about the Bartholemews?"
Pug walked toward her, stopped halfway there, then turned and sat down on the edge of the bed. "Petra," he said, "you know my mother is very close to Saffron Bartholemew. Our two families are...well, _close._ Over the years, my father and Whit's father have had converging interests in a number of matters. It could hardly have been otherwise. There have been times when the Ellisons were able to do favors for the Bartholemews, and vice versa. For Spirit's sake, Petra, that's how things _work_ in this galaxy! There's nothing underhanded or illicit about it; it's simply the way things get done."
"One hand washes the other?" Petra smirked.
_"Yes!"_ Pug ran his hand through his hair and breathed heavily. "Look, Petra," he said, "I don't expect you to understand this, because you haven't had the...the _background._ Among families like mine and Whit's, there are certain mutual relationships, certain understandings..."
"Which I couldn't possibly comprehend because of my... _background_?"
"I didn't mean it that way," Pug said. "Honestly, Petra, I'm just trying to explain why you should delete those files. Don't try to turn this into class warfare, all right? I know what you think of my family and my friends, but I'd appreciate it if you'd try to be a little more understanding. I mean, I put up with your mother, didn't I?"
Petra had no answer for that. She put the pad back on the dresser and went to sit down next to Pug.
"You're taking the Pelham job, aren't you?" It was more a statement of fact than an accusation.
"I haven't decided yet," he said.
"But your parents have. And that's all that matters, isn't it? Just another one of those understandings that families like yours have, right?"
Pug abruptly rose to his feet. "Give me a break, would you?" He turned and looked down at her. "You think this is easy for me? You think there aren't times when I want to tell my father to take his fucking empire and shove it?"
"You sound just like Whit Bartholemew," Petra observed. "And look what he wound up doing."
"If I wanted to run my father's business, would I have joined Dexta?"
"You might have," Petra replied. "Just a little youthful, pro forma rebellion, before returning to the fold. Isn't that the sort of thing that usually happens in families like yours?"
"You know, you're as big a snob as your mother. I don't hold it against you that you were born poor, Petra. Why must you hold it against me that I was born rich?"
Petra considered that for a moment and decided that it was a fair question. "Sorry," she said. "It's just that I need to know where we're going, Pug." She stood up, closed the distance between them, and put her arms around his neck. "I still want this to work."
"So do I." Pug drew her closer and kissed her softly on the neck. "And as for where we're going...yes, we're probably going to Pelham. Would that be so awful?"
Petra pulled back a little so she could look at his face. "I made a commitment to Gloria," she said. "We both did."
"And what if Gloria decides she'd rather be Empress? Where would that leave us?"
"I talked to her about it. She says she hasn't made up her mind. But if she did leave Dexta, I think she'd want me to come with her. Personal secretary, maybe."
"And I could be your assistant?" Pug released his hold on her. "Look, I really don't mind being your assistant now. But I don't want to spend my whole life doing that. This Pelham job could put me on the fast track at Dexta. If I rise high enough, soon enough..."
"Then you wouldn't have to come back here to the family business?"
Pug nodded. "That's right," he said. "In any case, if Gloria leaves, or OSI collapses, I wouldn't be hurt. And neither would you if you came along. There's nothing wrong with an Undersec slot on Pelham, you know."
"And in a year, when you make ImpeSec, I could be _your_ assistant."
"That bothers you?"
Petra sighed. "I guess not," she said. "Unlike you, it looks like I _am_ going to spend my whole life being _someone's_ assistant. I suppose that's the most I can expect, given the tragic limitations of my background." She offered Pug a weak smile, which he returned.
"Right now," he said, reaching into her robe and fondling her breasts, "I'd rather concentrate on your _fore_ ground."
"Which is also tragically limited."
"That's all right," he said. "I don't mind slumming."
CORNELL DUBRAY STARED AT GLORIA VANDEEN, chattering away about OSI on the far side of the room. If he was honest with himself—and he usually was, because he could afford to be—he had to admit that he admired her, and not simply for that magnificent ass, which was currently on full display in a bizarre bit of sartorial architecture that left her front mostly covered in swirling silver tinsel. VanDeen was smart and ambitious and, in her own way, as ruthless as DuBray himself. She was easily the most formidable woman he had encountered since Elsinore Chandra in her prime.
The Sector 21 Reception was well attended because it was the host Sector, and it was no surprise to find VanDeen here. Inevitably, he was going to keep tripping over her at these affairs throughout the length of the Quadrant Meeting. That could prove to be inconvenient; he didn't want any public scenes with her because there was no way he could come out on top in such an encounter. She had public support that he could never hope to muster. And, as she had pointed out in that broom closet this afternoon, Dexta people liked her. DuBray resolved to keep to the far side of the room from her and avoid confrontations.
That was ridiculous, of course, but there was no help for it. This was _his_ Quadrant, and—at least until Norman Mingus arrived—he was the senior Dexta official on New Cambridge. Yet that twenty-five-year-old harridan had somehow put him on the defensive.
DuBray helped himself to a fresh glass of champagne from a passing waiter and munched on a caviar-covered cracker. If he didn't watch it, he'd probably put on ten pounds in the next two weeks. Long ago, when he'd lived on this planet, he'd been lithe and athletic and didn't have to worry about such things. In bed, Saffron had called him "Slim."
Saffron was here tonight, too. They'd made their peace years ago—or an armistice, at any rate—but there were unexpected moments when he still felt a pang. It had all worked out for the best, in the long run, he supposed; fifty-five years certainly qualified as the long run. They had each prospered, the Empire survived, and history had buried all their tawdry little secrets.
Until now.
He was certain that he was correct in what he had said in the closet that afternoon. VanDeen truly had no idea what she was dealing with. She was like a dog tugging on the corner of some half-buried shroud because, just maybe, there was a bone somewhere down there. If he told her the truth, would she stop tugging, stop digging?
She might. But he simply couldn't risk it. Truth was like a plasma bomb; you only used it when there was no other choice. Truth, like unleashed plasma, was indiscriminate and unyielding and devoured everything in its path. Lies were safer because lies could be controlled. He had learned that, if nothing else, in seventy years at Dexta.
One of his aides caught his eye and signaled to him. DuBray nodded, set down his champagne, and followed the aide out of the mansion's ballroom, down a long corridor, and into a small, opulent private office. The aide retreated, closing the door behind him, leaving DuBray alone with the girl.
She stood before him, clearly frightened, but with her widely set dark eyes gleaming in anticipation. Something big was about to happen to her, and DuBray gained the impression that she might possibly welcome it.
"Ms. Murakami," he said, "thank you for coming. You know who I am?"
The girl nodded. "Yessir. Cornell DuBray, Quadrant 4 Administrator." Then she added, unnecessarily, "Level Four."
DuBray smiled. "That's right," he said. "And you are Level Fourteen, currently assigned to the Office of Strategic Intervention."
"Yessir."
She was very attractive, another one of OSI's young Tigers. She was wearing nothing on her small, slim body but a tiny scrap of black mesh band skirt and a matching shawl carelessly draped over her shoulders. Her face, Asian-Pacific, was carefully made up to emphasize her hypnotic dark eyes. She looked as if she might have been about twelve, but DuBray knew she was twenty-five. And he knew much more than that.
"Ms. Murakami," he said, "your father is currently incarcerated at the prison colony on Hingson III. He has two years left to serve on a three-year term for fraud and embezzlement. Would you like to get him out next month?"
Her eyes widened even more. "You could do that?"
"Of course I can. And I will, if you cooperate."
DuBray could tell from the way she hesitated that Murakami was not the innocent young waif she appeared to be. She was already calculating the angles, measuring the moment for maximum advantage, like a pool shark plotting a three-cushion shot. So much the better.
"I'll cooperate in any way I can, sir," she said at last.
"I know you will."
"What do you want me to do?"
"You are presently working as Gloria VanDeen's assistant, correct?"
She nodded. "Petra Nash is her regular assistant, but she's busy on another assignment, so I'm filling in for her."
"And just how close are you to Ms. VanDeen?"
The girl hesitated, then smiled slyly. "We've made love," she said.
"Have you, now?" DuBray found that intriguing, but not entirely relevant. "Does she confide in you? Tell you her plans? Share information?"
Murakami shrugged. "In general. I don't think she tells me everything that's going on, but I think I have a pretty good idea of what she's up to, most of the time."
"And how would you feel about sharing that knowledge with me?"
"You want me to spy on Gloria?"
"That's what I want. Will you do it?"
The hesitation again. The girl would not have been a good poker player.
"I...I'm not sure. I need to think about it."
"Think about _this,_ Ms. Murakami. Your father is in prison on Hingson III. I can have him released, or I can send you there to join him."
A look of sudden doubt flickered across the girl's features.
"On an irregular basis for the past two years," DuBray said, "you have been involved in the sale and distribution of illegal substances—Orgastria-29, to be precise."
"How did you know that?"
"How I know is of no moment. What matters is that I _do_ know. Well, Ms. Murakami, can I count on your cooperation?"
"I'll do it," she said.
"Of course you will. After we're through here, my aide will instruct you in your duties and communication procedures."
"How do I know I can trust you?" Murakami asked. She had some nerve, DuBray was pleased to see.
"If you do as I ask, your father will be released. Frankly, I don't care a farthing whether your father is released or rots in prison. I have no interest in the matter, so I have no reason to lie to you."
"And what about me?"
"What about you?"
"Naturally, I want to see my father released. But I could be risking my Dexta career by doing this for you, Mr. DuBray. I mean, what's in this for _me_?" She gave him that sly smile again.
DuBray smiled back at her. "I'm glad you asked me that," he said. "Selflessness makes me suspicious. Very well, then, what's in it for you is a promotion to Thirteen when we return to Earth and a transfer to an appropriate position on Quadrant staff. Does that meet with your approval, Ms. Murakami?"
"Yes...but wouldn't that let Gloria know what I've been doing?"
"By then," DuBray assured her, "it won't matter."
EDWIN OGBURN, PRESIDENT OF THE REPUBLIC of New Cambridge, made the introduction. "Ms. Gloria VanDeen of Dexta, allow me to present Ms. Saffron Bartholemew, one of our world's most beautiful and respected citizens."
They shook hands politely and took each other's measure. Saffron's resemblance to her father was unmistakable; something in the aristocratic slope of her nose and the high, unlined forehead. Her fine, cornsilk hair was artfully arranged, and her V-shaped jaw jutted out just a little. Her eyes were the same color as her father's, a watery blue-gray.
Both women simultaneously turned their heads to look at the man who had introduced them. Ogburn, no fool, quickly made his apologies and departed, leaving the two women alone.
"So," Saffron said at last, "you're my father's latest popsy."
Gloria ignored the verbal sally and smiled pleasantly. "I'm hardly that," she said, "but I do work closely with him."
"All the stories and rumors aren't true, then?"
"I doubt if one percent of them are true. But we aren't intimate, if that's what you're getting at."
"No?" Saffron frowned, as if disappointed at this news. "Well, he is a hundred and thirty-one. Finally slowing down, I guess."
"I wouldn't know about that. But the fact is, his health has not been good of late. He just had a new pancreas put in. He'll be here next week, Ms. Bartholemew. Perhaps you should take the opportunity to find out for yourself how he is, instead of relying on rumors and secondhand reports."
Saffron raised an eyebrow. "I'll take it under advisement," she said.
"He's a lonely old man," Gloria said. "It would do him good to see his family."
"Ah, but would it do his family any good?"
"I don't see how it could do you any harm."
"Then, Ms. VanDeen, you don't know my father as well as you think you do."
"Look, Ms. Bartholemew," Gloria said, feeling a growing sense of exasperation, "I don't know anything at all about Norman's personal life. I don't know why the two of you are estranged, and I don't want to know. But I think he's a good man, and I care very much about him."
Saffron gave her a lingering, appraising stare, like a jeweler considering a gem. Then she said, "You really do care about him, don't you?"
Gloria nodded. "I do."
"No reason you shouldn't, I suppose," Saffron said. "He does have...qualities. You know, I was just about your age when we became, as you say, estranged. Until that point, I worshipped him. Thought the galaxy revolved around him. And I suppose it does, now. Even then, you could sense greatness in him. But greatness is not necessarily an endearing trait, Ms. VanDeen."
"Is that why you resent him? Was he too busy being great to be a good father?"
"Not at all. I wasn't neglected—far from it. My mother died in an accident when I was only twelve, you know. And my father was very devoted to me. Credit where credit is due—I can't deny him that. He positively doted on me, and took great pains to see to it that I grew up with every advantage. He was also demanding, but not oppressively so. I think he simply wanted me to be perfect so that I could have a perfect life."
"And now you blame him because you didn't?"
Saffron didn't quite laugh at that. "That's one way of looking at it," she said. "Ms. VanDeen, you are obviously unaware of what transpired between my father and me. Perhaps my father will see fit to enlighten you someday, but I have no intention of dredging up painful memories just for your sake. I'm glad that I had the chance to meet you, Ms. VanDeen. Allow me to give you some advice. Whatever affection or esteem you feel for my father, and whatever his emotional attachment to you may be, don't for one minute imagine that you are anything more to him than what I was—what everyone is."
"And what is that?"
"A tool, Ms. VanDeen. A tool."
PETRA AND PUG RETURNED TO THE OLD ANNEX the next morning to continue their research. They had said no more about the files in her pad, but when Petra connected the pad to the main console, Pug gave her a hard look.
"You're going to do this in spite of everything?"
"It's my job," she told him. "Yours, too, or had you forgotten?"
Pug snorted wordlessly and got to his feet. "I'll be over in Gibraltar," he said. "I'm going to see if I can find some of those people who were here in 3163."
"Good idea. Maybe they'll even talk to you."
Pug left, and Petra finished downloading the files from her pad. After all the talk, she was eager to get down to cases and see what was in them. The main thing she was after was a bill of lading for that final shipment to Savoy. Aside from the 24,000 Mark IV plasma rifles, they still didn't know what else was in that shipment. Whatever it was, it was now probably in the possession of PAIN.
She quickly zeroed in on August and September of 3163. Before she found the bill of lading, she happened upon the log and registration information for the freighter that had been used for the shipment. She made note of the names of the captain and crew of twelve, thinking that some of them might very well still be alive and willing to talk, assuming they could be found. Officially, at least, they had all disappeared, along with their freighter, when the Ch'gnth attacked. The freighter itself, a LoadStar, was rated at sixty thousand tons capacity and was seven years old in 3163. B & Q had bought it new, for 73 million crowns, with a loan from the Bank of New Cambridge. Before its final voyage, the vessel had logged just over thirty-nine thousand hours in Yao Space and 42,154 light-years traveled. There was nothing interesting or unusual about it, as far as Petra could tell.
Next, she studied the contracts that had been negotiated between Dexta and B & Q. Whitney Bartholemew had signed for B & Q; as Quincannon said, it had been "Bart's deal." There were a number of signatures for Dexta, as required by a web of regulations: everyone from Port Masters to the Export Decontamination Control Authority to Assistant Quadrant Administrator Cornell DuBray had been required to sign it, and did. Petra stared at the contracts for quite some time. She had already seen the Dexta copies of the same documents. Out of curiosity, she called them up and did a side-by-side comparison.
There were differences between them. Nothing dramatic, or even meaningful, as far as she could tell. The standard wording in the contract text was nearly the same, but the fill-in sections of the documents contained minor variations in style and language, almost as if they had been completed at different times by different people. That, in itself, was not particular cause for concern. It was not unreasonable to assume that in the back-and-forth of the transaction, different people at Dexta and B & Q might have gotten their hands on the contracts. Still, now that she knew the truth about Whitney Bartholemew, Senior, Petra couldn't help wondering.
She got to her feet and paced around the tiny office space they had been granted. For today, she had shed her Tiger stripes and was wearing jeans, a loose gray shirt, and, praise the Spirit, comfortable shoes.
If something was wrong with the contracts, how could she tell? How could she recognize something for what it was, even if it was staring her in the face? Answer: she couldn't, because she had no idea what contracts like this were supposed to look like in 3163. But there was a cure for that gap in her knowledge.
With a sigh of resignation, Petra sat down again and called up a random assortment of B & Q and Dexta shipping contracts from 3163. She studied them for a full hour before returning to the Savoy contract. This time around, the truth leaped off the pages at her. She double-checked, and there was no doubt.
The Savoy contract in the B & Q records had been written, not in late August 3163, but no earlier than late October of that year. With the coming of war, in early September, the basic shipping contract text had been altered to include war-related issues. In the newer version of the contracts, there were additional lines and boxes, and some tacked-on paragraphs related to wartime liability, insurance riders, military escort regulations, and the like. None of that new material appeared verbatim in the Savoy contract, and yet Petra had the feeling that it had been there, then deleted. The result was that the spacing and some of the phrasing in the Savoy contract was subtly different from other prewar contracts.
But what did it mean?
The Savoy contract had been written—or rewritten—after the fact, and imperfectly altered to make it appear as if it had been written in August. Why?
No answer occurred to her.
Finally, she put the contracts aside and resumed her search for the bill of lading. When she found it, she gave it a once-over, without really registering any of the information it contained. She moved it to a corner of her screen and called up another random assortment of bills of lading from the relevant period. After another hour of study, it was clear to her that the same thing that had been done to the contracts had been done to the bill of lading. Again, why?
She returned to the Savoy shipment bill of lading and examined the specific contents. As expected, it listed the 24,000 Mark IV plasma rifles, gross weight, 264 metric tons. A Mark IV, she calculated, weighed—with packing material—about eleven kilos. No wonder they phased them out.
Along with the rifles, there had been 2.4 million plasma cylinders—extra ammo, in other words. Gross weight, 2400 tons.
Ten thousand cases of plasma grenades. Gross weight, 1200 tons.
What else? Three point six million Imperial Marine Corps Field Ration packets. Gross weight, 2160 tons.
Two hundred and eighty-six orbital plasma mines. Another 5148 tons.
And—holy shit!—eighteen 200-quadrijoule plasma bombs. Ninety-five lethal tons. PAIN had plasma bombs!
Petra had to sit and think about that one for a while. So far, PAIN had been content to shoot up Dexta offices. They could have been blowing up entire cities!
She was shocked and frightened by the discovery. But there was nothing she could do about it, so she returned to the bill of lading and continued adding up the monstrous inventory.
Except that it _didn't_ add up.
JILL CLYMER, WAITING OUTSIDE THE DOORS TO Wendover Freight and Storage in downtown Central, saw Eli Opatnu approaching and felt her heart skip a beat. Did his pheromones work at a distance, she wondered, or was she just reacting to the man himself? She had always thought of herself as a pretty cool customer when it came to the male of the species. She enjoyed sex and had never been shy about it, but until now she had always felt able to keep a rein on her emotions. Her ex-husband had even complained about it. She never let herself go, he said, never abandoned herself to the raw realities of sex. "You think you can wade across a raging river and never get your feet wet," he had said.
And yet, here she was today, doing her best Gloria VanDeen impression in a nearly transparent shirtdress, unbuttoned and flapping open to reveal her breasts and thighs, not a stitch on underneath, damn near as naked as Petra had been the other night. And all for Eli Opatnu, a man who collected women the way some people collected butterflies.
And here he was, damn him, smiling to beat the band, eyes gleaming, supple hips and broad shoulders moving rhythmically as he walked toward her. So utterly sure of himself, so full of life, so insidious. Jill took a deep breath and returned his smile.
"Sorry I'm late," he said as he clasped both of her hands in his. "Hard to get a cab this week." _His hands are warm; or maybe mine are just cold,_ she thought.
They went inside and introduced themselves to the woman at the main desk. Jill flashed her Dexta ID, but the woman scarcely noticed; her attention was focused on Eli. He gave her his best smile, and within a minute, they were ushered into the office of Wendover's regional manager, a Ms. Sophie Darceau. _Swell,_ Jill thought.
Ms. Darceau didn't quite ignore Jill, but seemed to regard her as a distraction from the main business of the moment, which was making small talk with Eli Opatnu. Finally, Jill simply interrupted them and held up her pad. "Ms. Darceau," she said, "I have here a list of nine freighters owned or operated by Wendover in this Sector. Some discrepancies concerning their records have turned up, and we would like to clear up the confusion. I'll transmit the list to your console, and I would appreciate it if you could give us a full accounting of the whereabouts of those freighters for the last two years."
Ms. Darceau looked up, as if noticing Jill for the first time. "What? Oh, yes, I see. Now that I think about it, we did receive a memo concerning this from the home office on Staghorn."
"Then you know that Dexta is conducting a formal investigation."
"Yes, that's what they said. But they also said that I was not to give out any proprietary information without first consulting Staghorn."
"That would take weeks," Jill pointed out.
"Yes," said Ms. Darceau, "I suppose it would."
"Ms. Darceau," Jill said with more patience than she felt, "if you make it necessary, I can go see an Imperial Judge and be back here this afternoon with a warrant."
"I'll consult with our legal department, in that case. My understanding is that we can challenge the warrant."
"But why go to all that trouble?" Opatnu broke in, smiling ingratiatingly. "Look, Ms. Darceau, Jim Takahashi—your boss on Staghorn—is an old friend of mine. I know he'd want you to cooperate. In fact, as soon as I return to Staghorn, I'll tell him exactly what happened here and that you made every effort to comply with the memo from the home office. I know he wouldn't want a lot of unnecessary legal entanglements...warrants, suits, countersuits, that sort of thing. And _you_ wouldn't want that, would you?" He turned up the smile to maximum wattage.
"Well..." said Ms. Darceau, "I'd still have to consult with Legal."
"Of course. We understand that. Why don't you do that, then, and we'll be back this afternoon to see where we stand. I'm sure we can work this out without dragging the courts into it." He got to his feet and held out his hand to Ms. Darceau. "A very real pleasure meeting you, Sophie. Have a pleasant day, and I'll see you this afternoon."
Jill followed him out of the office. On the sidewalk, Opatnu turned to her and grinned. "There! Now we have two or three hours for a leisurely lunch."
"But we don't have the data on those freighters."
"We'll have it this afternoon. If we get a warrant, we'll just wind up in a legal hassle that might take weeks. Trust me, we'll get what we need. Now, what about lunch? There are some lovely restaurants down on the waterfront overlooking the strait. Shall we?"
They settled in at an outdoor table on a terrace two hundred feet above the choppy blue waters of the strait. Jill noted that Opatnu was careful to sit upwind from her. Was that so she would get a full dose of his pheromones, or merely to get a better view of the breeze flapping around her unbuttoned shirt and baring her breasts? Both, probably; Jill had already noticed that nearly everything Opatnu did or said was carefully calculated.
The man annoyed her and fascinated her in equal measure. That, in itself, was annoying. Why couldn't she get a handle on the situation? Why did he make her feel so unsure of herself, so helpless and foolish? Like some damned adolescent.
Opatnu asked her about her past and her homeworld. She told him that she was from Minnetonka, an agricultural planet in Sector 2, five hundred light-years from Earth. It was something of a backwater, rural in its economy and attitudes, and Jill had grown up longing for bright lights and big cities. She had gotten a taste of that early on when she accompanied her father to sessions of Parliament on Luna. Jill found herself telling Opatnu about those trips, and about her father, a man she had loved and admired without limit.
"Yes," Opatnu said, "I vaguely remember the story. He caused a bit of a stir in Parliament ten or fifteen years back, didn't he?"
Jill smiled, then pointed to a swirling eddy just offshore in the strait. "See those waves?" she asked. "That was my father. A tiny ripple in a big, blue sea. In about a million years, that ripple will probably carve out a bit of the shoreline, just through sheer persistence. My father thought that if he persisted, he might get a few worthwhile things done, eventually. But they didn't give him the time he needed."
"There was some sort of scandal, wasn't there?"
Jill nodded. "He found some corruption and tried to expose it. Oh, it wasn't anything major, just some illegal loans that a Parliament committee was dishing out to some cronies back in Sector 2. The kind of thing that goes on all the time, I suppose, but it offended my father. He thought that the money going to those big shots should have been going to people who actually needed it—his constituents, the people who voted for him and sent him to Parliament to represent them and protect their interests. Dean Clymer was a man who took his responsibilities seriously."
"And what happened?"
"The big shots back home didn't appreciate what he was doing. So they dug up something from his past and used it to smear him. Again, it was nothing major. But the thing was, it was true. My father made a mistake thirty years ago, but thought that he had made up for it by being an honest and dedicated public servant. When the mistake was dredged up, people began to wonder if Dean Clymer was really the honest man he seemed to be. And my father couldn't bear the shame of it. So he killed himself."
Jill finished the story and turned to look out to sea, so she wouldn't have to look at Opatnu. So he wouldn't see the tears welling in her eyes.
Opatnu reached across the table and clasped his hand around hers. "I'm glad you told me," he said. "I wondered why a measly little double-flagging scam seemed to be so important to you. Now, I think I understand. It isn't about Wendover, is it? It's about your father."
Jill blinked back her tears and nodded. "I think of him all the time," she said. "I imagine him looking over my shoulder, asking me if I'm sure I'm doing the right thing. That was one reason why I was glad when Gloria asked me to join OSI. It seemed like a way to make a career out of doing the right thing." She smiled self-consciously. "Silly, huh?"
"Not at all. It's admirable. But I get the impression that you have some doubts."
"Not doubts, really. Gloria wants to do the right thing, too. Her style is a little overwhelming at times, but she gets the job done. But this whole business of turning OSI into the Fifth Quadrant...I don't know. I'm afraid she's getting too caught up in bureaucratic rivalries and internal Dexta politics."
"How could she avoid it?" Opatnu wondered.
"I know, I know. That's the game at Dexta, and she has to play it. But I can't help worrying that she's making the same kinds of mistakes that my father made, and that eventually she'll have to pay for it. You start out making little compromises to serve some higher purpose, and you wind up making bigger and bigger compromises, because, hey, you've _already_ compromised yourself, right?"
Opatnu frowned. "It sounds like you're saying that any compromise at all is fatal. That's not a very realistic attitude."
Jill gave Opatnu a rueful smile. "That's us Clymers," she said. "Pure but stupid."
"I admire you for your purity, Jill, as well as for your stupidity. Maybe we could all benefit by a little more of that kind of stupidity."
"My father used to say, 'When you sup with the Devil, you need a long spoon.' "
Opatnu released his hold on Jill's hand and picked up one of her spoons from the table setting. He examined it carefully and said, "Looks about the right size to me."
"We'll see," she said.
WHEN PUG DIDN'T RETURN, PETRA WORKED straight through the lunch hour and into the afternoon. Finally, hunger and curiosity drove her out of the Old Annex and into the bustling streets of Central. It was only a mile or so to the Old City and the office of B & Q Shipping, so she decided to walk. She undid a couple more buttons on her shirt, figuring she would give lecherous old Jamie Quincannon a thrill. And herself, too. She couldn't deny it—this Tiger business was getting to her. Whit Bartholemew had said she was the sexiest woman he'd ever met, and even if he had been lying, it was still nice to hear.
She stopped in a small shop for a sandwich and coffee. Cream, but no sugar; she had noticed that her jeans seemed a little tight. Too much high living, not enough Qatsima lately. The walk would do her good and give her a chance to think.
Contracts, bills of lading, and gross weights swirled through her mind as she walked. She wondered if she was making too much out of what she had seen. In her ignorance and suspicion, could she be tilting at a windmill of her own construction? If she hadn't known that Whit Bartholemew's father was a zamie, would she have made anything at all out of the minor anomalies she had uncovered? Did they even mean anything, or were they just the routine slips of harried bureaucrats and flawed human beings? She remembered a history professor in college who had said, "If you hear hoofbeats, don't go looking for zebras." He had also said, "Incompetence explains more of history than conspiracies."
And yet, this conspiracy was real. Weapons from the Savoy shipment had undeniably turned up in the hands of PAIN terrorists. But how far did the conspiracy go, and what did it all mean? The records she had downloaded were sterile, lifeless. They listed facts—or someone's preferred version of the facts—but they didn't tell her what had really been going on fifty-five years ago. Maybe Jamie Quincannon could tell her.
Maybe he could tell her why 27,542 metric tons of cargo had been shipped to Savoy on a freighter that could hold 60,000 tons.
It made no sense. Why send a valuable, half-empty freighter off on a risky errand to a place where war might erupt at any moment? Even if the freighter had never gone to Savoy, why send it _anywhere_ half-empty? Petra knew that the economics of interstellar trade were exacting and unforgiving. The expense of traveling between the stars was simply too great to waste cargo space. She and Gloria usually traveled via a Flyer, which was nothing but a tiny tin can with an engine attached, yet even a Flyer trip was expensive. Sending a sixty-thousand-ton freighter on a seventy-five-light-year hop to Savoy without a full hold would be not only stupid, but unprofitable. Why not send the shipment in a smaller freighter? That would be cheaper and faster.
Maybe no smaller freighter had been available. That was something she could check in the B & Q records. But then, why use B & Q? Why not let the contract to a big outfit, like Trans-Empire, that would certainly have an appropriate ship available? Or would Trans-Empire have been reluctant to send a ship into a potential war zone?
And what was Bartholemew's angle? An insurance scam? Possibly...Quincannon had said that his partner knew how to play both ends against the middle, so maybe that was it. B & Q had collected on the insurance a few years later; why collect for a small ship when you could collect for a big one?
Meanwhile...what had happened to the shipment itself? However it was shipped, and whatever angles people were playing, it had gone _somewhere._ And wherever that place was, PAIN had found it.
If anyone knew the answers, it would be Jamie Quincannon. And Petra had a feeling that, with the proper inducement, he just might tell her what she needed to know. She undid one more button, then entered the building, and, with a game smile fixed on her face, assaulted the five flights of stairs.
It was dark at the top of the stairs, no light at all except for what filtered in through a couple of grimy windows. Petra walked slowly over the creaking wooden floor and waited for her eyes to adjust to the gloom. Her left foot suddenly skidded ahead, and she nearly fell. Gathering herself, she stooped down to see what had caused the skid and noticed a dark liquid staining the floor.
Rising, she looked into an open room on her left and saw the source of the dark liquid. It was the body of Jamie Quincannon.
GLORIA GATHERED HER TROOPS IN THE SUITE at her hotel the next morning. "I thought it would be a good idea to bring everyone up to date on recent developments," she said. "First off, I want to thank you again for the great job you've been doing. From the feedback I've been getting, I think we've done ourselves a lot of good here this week. Everyone's talking about OSI, and most of the comments I've heard have been very positive."
"I should _hope_ so," Althea Dante cackled. "I've devoted body and soul to the cause."
"I won't ask in what proportion," Gloria said. "Now, I'd like to announce that our very own Elaine Murakami will be promoted to Thirteen when we return to Manhattan. In the meantime, she will be spying on us for Cornell DuBray, so if there's anything you don't want the Quad Admin to know, don't say it around Elaine."
Elaine grinned happily. "He asked me just the way you said he would," she said. "How did you know?"
"Lucky guess," Gloria replied. She didn't add that it was what she would have done in DuBray's place. "We'll feed him just enough information to keep him happy and make him think he knows what we're doing. Now, Jill, I understand that we've run into a snag on the double-flagging investigation."
"I'm afraid so," Jill said, frowning. "Eli Opatnu and I went to the local Wendover office yesterday morning. I thought we were going to get what we needed from them without any hassles. But when we returned in the afternoon, we found that while we were at lunch, Wendover had gone to court and gotten a restraining order against us. We can't file any subpoenas or get any warrants until a lawyer from their home office arrives here to file a brief. That will be at least a month." Jill looked disgusted.
"Is there any chance that we can overturn that ruling?"
"Possibly," Jill said. "But that would mean bringing in the local Dexta office for Sector 21, and since one of the things we're investigating is whether there is any Dexta involvement, that could defeat our purpose. Of course, by now, they've heard about the restraining order, so if anyone here is involved in the double-flagging, they've already been alerted."
"I see," said Gloria. "So what are our options?"
"I can stay here for a month and try to see this through on my own. Or, when we get back to Earth, we can send one of OSI's lawyers here to deal with it. Either way, we're stalled for at least a month. No recent word from the Financial team we sent to Staghorn, so I don't know where we stand at that end."
"Well," Gloria said, "let me contact the Sector 21 Administrator and see what I can do. In the meantime, you might as well join Althea and the boys on the committee circuit and start schmoozing."
Gloria turned to Petra, who was sitting on a sofa next to Pug, glumly staring off into space. Petra had spent most of yesterday afternoon dealing with the Central Police. She had called them when she found Quincannon's body, not knowing what else to do. When the initial shock had worn off, she had called Gloria, who dispatched Arkady Volkonski to the scene. The Central Police, who had had their fill of Dexta Internal Security in the past week, tried to keep him out of the way while they interrogated Petra, but he had simply run roughshod over them and rescued Petra from a grim ordeal. Volkonski, citing the superior mandate of Dexta and Imperial Security, had told the cops as little as possible about Petra's interest in Quincannon and B & Q Shipping, then marched out of the building with Petra in his grip.
"Petra," Gloria said, "I know you had a rough day. But is there anything you can tell the rest of us about your investigation?"
Petra looked around the room. She was among friends, but she seemed reluctant to talk about what had happened. "I've turned up a few things," she said. "I'm not really sure what to make of it all. But, Arkady, there is something that you should know. The bill of lading for that Savoy shipment mentioned eighteen two-hundred-quadrijoule plasma bombs."
Volkonski nodded. "Internal Security suspected as much, but it's good to have confirmation. I'll inform Quadrant Security immediately."
Gloria stared at Petra for another moment, then moved on to Pug. "How about you, Pug? Anything?"
He shook his head. "I found a couple of people yesterday who were here in 3163, but they didn't know anything about that shipment. There are three others that I couldn't find, but they are supposed to be around here, somewhere. I'll keep trying."
"Okay." Gloria got to her feet and the others did the same. "I want all of you to stay alert. Arkady's Bugs will be assigned to cover each of us, just in case, although I doubt that what happened yesterday has any direct connection to OSI. So let's all just do our jobs and be happy in our work. Have a good day, people."
Jill, Althea, Brent, Darren, and Arkady left the suite. "Petra," Gloria said to her, "would you stay for a few minutes? Pug, she'll catch up to you later." Pug nodded and left.
"Do you want me to stay, too, Gloria?" Elaine asked.
"That won't be necessary," Gloria told her. "I want to have a few words alone with Petra."
"Are you sure?" Elaine asked again. "It's no trouble, really."
"Don't take this spying business too seriously, okay? I'll see you in a while at the Convention Center," Gloria said. Elaine walked slowly from the room. When she was gone, Gloria went to Petra and hugged her. "I'm awfully sorry about what happened. It must have been a terrible thing for you."
Petra shook her head. "I was just stunned," she said. "I don't think it really got to me until this morning. I woke up and it suddenly hit me that Jamie Quincannon _wouldn't_ be getting up this morning. Nice old man, must have been a hundred and twenty, and suddenly, it's all just... _over._ Spirit, Gloria, I feel awful! I mean, what if something I did...?"
"None of that," Gloria insisted. "You can't let yourself think that way. You have a job to do, and a lot of lives might depend on your doing it right. Quincannon's death doesn't change anything, Petra. You can't start second-guessing yourself."
They sat down on the sofa and Gloria poured coffee for them. Petra took a small sip of hers, then put her cup down on the coffee table. "Gloria?" she said. "There are some things...things I didn't tell the police."
"Don't worry about it. You did the right thing. Quincannon's murder may have had nothing to do with our investigation. Let the police handle their end of it, and if anything turns up to connect it to what we're doing. I'll inform them when I think it's appropriate."
"But there are other things, too. Spirit, Gloria, this is turning into something that I didn't...didn't expect. Did you know that Whitney Bartholemew, Senior—"
"Was in the zamitat?"
Petra's mouth fell open. "How did...?"
"I have some additional sources of information," Gloria told her. The additional source had been Anton Grosz, who had delivered a one-time-only message from Ed Smith the previous evening. Gloria had played it on her pad; then the message destroyed itself and any traces of its presence.
Smith had told her that he had talked with some "senior people" in his organization. He had learned that Bartholemew had "scammed the insurance" on the missing freighter, making a handsome profit, while selling off the freighter that had presumably been lost at Savoy. Meanwhile, Smith said, the original Savoy consignment had been spirited away from New Cambridge on three separate shiploads. His sources didn't know where it had all gone.
Petra seemed annoyed that Gloria was ahead of her. "You mean I've been digging through all those old files, and you already knew everything?"
"Not at all," Gloria assured her. "My source has only very sketchy knowledge of what happened. But he did say that the shipment that was supposed to have gone to Savoy was broken up into three different shipments and sent somewhere from New Cambridge. He didn't know where."
"Three?"
"That's right. Why?"
"That doesn't make sense. I mean, one of the weird things I discovered was that the Savoy shipment only came to about 27,000 tons, and the freighter had a capacity of 60,000 tons. It would have been flying half-empty. But now, you say those 27,000 tons were divided in three, then sent away in other freighters."
"Maybe they were smaller freighters," Gloria suggested. "Three ten-thousand-tonners."
"Yeah," Petra agreed, "that would explain it. But that really complicates things."
"How so?"
"Well, a 60,000-ton freighter is strictly space-to-space. They can't land. But a ten-thousand-ton freighter can make planetfall."
"So?"
"I was thinking that we might have been able to trace where that original freighter went instead of Savoy. There are only so many planets that have orbital storage facilities. We could have checked every appropriate planet within, say, two hundred light-years of New Cambridge to see what 60,000-ton freighters docked at their orbital stations during, say, the last four months of 3163. It would have been a huge job, but eventually we might have turned up something. But now, if they used three smaller, planet-capable freighters..."
"I see what you mean," Gloria said. "They could have gone practically anywhere. But still, that was a good idea, Petra. And something else occurs to me. Did the B & Q records say anything about what _other_ freighters B & Q had available? Like, say, some ten-thousand-ton jobs?"
Petra's face lit up. "I didn't check for that, but I will. Wow, Gloria, that's a great idea! I'll get right on it."
"That's why we pay you the big bucks, Petra. Sounds like you've been making a lot of progress."
Petra shook her head. "I'm not sure if I have or not. It's all pretty confusing. One thing, I think that the Dexta contracts with B & Q were altered, or written after the fact." Petra explained in detail the reason for her suspicions.
"Interesting," Gloria said. "And you say that Cornell DuBray signed the contracts?"
"DuBray and a lot of other people. Don't make too much out of that, Gloria. It would have been odd if DuBray or someone in his office _hadn't_ signed those contracts."
"I suppose so." Gloria sighed. "Damn, if we could just tie him into all of this, somehow..."
Petra looked down at her coffee cup. "We can," she said.
"What?" Gloria stared, wide-eyed, at her assistant. "Out with it, Petra! What have you got?"
Petra sighed heavily, then looked at Gloria. "I had lunch...I mean, I met with Whitney Bartholemew, Junior two days ago. I learned some things...personal things...that sort of connect DuBray with Whit's father. I'm not sure exactly how, but...well, it seems that DuBray was engaged to Saffron Mingus at about the same time we're investigating. And then, Whit says that Norman Mingus forced them to break off the engagement and sort of delivered Saffron into the arms of his father. I don't know what it all means, but if you're looking for a connection..."
"That certainly qualifies," Gloria said, leaning back against the sofa. "Spirit...you say _Norman_ did all of this?"
"That's what Whit says. Of course, he hates his grandfather, so who knows what the real story is? Except that I think he was telling me the truth."
"I think so, too," Gloria said. "I met his mother the other night. Did Whit say why he did it?"
"Just that it suited his needs. It was all about power, according to Whit. He's kind of hung up on that subject. Gloria? There's something else you need to know about all of this. I didn't just have lunch with Whit. I made love with him."
Gloria looked at her. Petra didn't exactly look guilty, but she didn't look very happy, either.
"I see," Gloria said slowly. "And do you think that Whit could have been saying all these things just to get you to go to bed with him?"
"I suppose it's possible. That's why I thought I should mention it. I _think_ he was telling the truth, and I don't think he had any reason to lie. I mean, it was pretty obvious from the moment I walked in the door that we were going to wind up in the sack." Petra practically blushed, not in guilt but in embarrassment, it seemed to Gloria.
Petra sighed. "Maybe I'm just not cut out to be a Tiger," she said.
"Petra, you don't have to be _me,_ you know. I don't need a clone for an assistant. I need _you_...Petra Nash. That cute little girl from Weehawken...remember her?"
"Vaguely," Petra said. "But she's been running with some pretty fast company lately."
"You think she got lost?"
"Maybe. Dammit, Gloria, I just don't know anymore. I mean, I enjoy all the outrageous things I've been doing, and wearing these ridiculous outfits and, yes, I enjoyed being with Whit. Pug's been cheating on me with Steffany Fairchild—at least, I _think_ he has—and I just figured...well, you know. But so much has been happening, with Pug, and the Ellisons, and Whit, and this whole investigation, I just don't think I'm sure who I am anymore, or where I'm going." She looked at her boss. "And speaking of where I'm going, I think Pug's going to take that job on Pelham."
"I see. And are you going to go with him?"
Petra started to answer, stopped, started again, then simply threw her hands into the air. "Who knows?" she said with a forlorn laugh.
"Well, I'm in the same boat, kiddo. I don't know what the hell I'm going to do about Charles. He's coming here, by the way. Going to make some major address at the end of the Meeting. I just hope he's not planning to announce that we're getting married again."
"Whit said something to me about those ancient princesses that used to get married off to royalty because of power politics. He says that's pretty much what happened to his mother. Gloria? Don't let it happen to you."
Gloria chuckled, then squeezed Petra's knee. "Thanks for the advice. But it's not my family that's putting the pressure on. It's me. Oh, my mom would love it if I became Empress, but the main thing is that I think _I_ might love it, horrible as that sounds." Gloria got to her feet and struck a regal pose. "The Empress Gloria, Mistress of All She Surveys. Got a ring to it, don't you think?"
"You'd be great at it, Your Highness—if that's what you really want."
Gloria flopped down on the sofa again. "That's the kicker, isn't it? Petra, how does _anybody_ know what they really want? I want to run Dexta someday, but I also sort of want to be Empress."
"And I want to stay with Pug—but I also want to stay with _you,_ whether it's in Dexta or Rio or wherever. It's not always easy being your assistant, Gloria, but it's never been boring."
Petra's brow suddenly furrowed, and she looked very serious. Gloria looked at her and said, "What?"
"Gloria? They beat him to death. I mean, someone pounded on his face until it looked like strawberry jam. At first, I wasn't even sure it was him. I had to take a good, long look. I don't think I'll ever forget that. I want to ask, who would do a thing like that? But I already _know_ who would do it, and what's more, I know why. The zamitat."
"We can't be sure of that," Gloria said.
"It was them," Petra insisted. "Gloria, there are some things I haven't mentioned because...well, because it might get both Pug and Whit in trouble. If I tell you, you have to promise me you won't do anything about it."
Gloria was silent for several moments. "That's a lot to promise," she said at last.
"I know. But you have to promise."
"All right," Gloria said. "You have my word."
Petra took a deep breath. "Jamie Quincannon let me download all of B & Q's records the minute I asked. He didn't really care. But he made me promise not to tell Whit Junior. But Whit found out anyway. Apparently it showed up on his master computer. Anyway, because of his father's zamitat connection—and Whit says that _he_ isn't connected with them—well, he said that the stuff in those files could be embarrassing and damaging. And, well, he asked me to destroy them. I told him it was a felony just to ask me to do that, but he kept on asking."
"A natural enough thing for him to ask, under the circumstances," Gloria said.
"But that's not all. That night, Pug asked me the same thing. He said his family and the Bartholemews had always been close and had done a lot of favors for each other. And so he asked me to destroy those files. And both of them, Whit and Pug, said that it could be dangerous if the zamitat ever found out what I had. What Quincannon had done."
"But how could they have found out?" Gloria wondered.
"Maybe the same way Whit found out. Maybe they've got a monitor on all of Bartholemew's old computers. Who knows? But Gloria, who else could have done it? And why? It _had_ to have been the zamies."
"Maybe," Gloria said. "In any case, I'm going to order extra security for you."
"But you won't do anything about Whit and Pug, will you?"
"I said I wouldn't. I can understand Bartholemew. But I'm disappointed that Pug would ask that. He knows better."
"I think he felt it was something he owed his family, and Whit's. You're rich, Gloria. You must understand these things better than I do."
Gloria shrugged. "I suppose so," she said. "But Petra, I'll tell you this. I'm proud of you for not destroying those files."
Petra gave her a wan smile and stood up. "Then I suppose I should get back to them, shouldn't I? Who knows _what_ might turn up?"
THE FIRST TERRORIST ATTACK HAPPENED THAT night.
It was a hit-and-run raid, featuring plasma pistols and chemical bombs that made a lot of noise but didn't do a great deal of damage. The target was a reception at the K'Spanci consulate. The K'Spanci were a race of flightless, owl-like creatures who had been members of the Empire for more than two centuries. Like other nonhuman species, they felt left out and neglected at affairs like the Quadrant Meeting. They complained—and not without good reason—that extraterrestrial species were underrepresented in Dexta, and that the Empire unfairly discriminated against them in favor of _Homo sapiens._ The K'Spanci reception attracted a fair number of alien representatives, but only a smattering of humans. That made it an attractive target for the terrorists—not PAIN, but PHAP, the Pan-Human Alliance for Purity.
The attack was a failure, in that no aliens, but two humans, were killed. So were two of the three terrorists. The third was captured by Internal Security, who quickly extracted enough information from him to be able to stage a raid on the PHAP command post, such as it was, where two more terrorists were taken into custody.
It was a relatively minor event, but it served to put everyone's nerves on edge and caused security to be ratcheted up another notch. Gloria noticed the heightened tension as she made the rounds of committee meetings the next day, and receptions the following night. She was feeling it, herself, and decided to do something about it. The next morning, she rounded up her troops and declared that Sunday would be a mandatory day off for everyone in OSI. Just to be certain that her orders were carried out, she organized a group excursion to a beach resort on the south coast of the continent, and attendance was definitely _not_ optional. Her people were going to relax if it killed them.
Gloria herself was late arriving from the Transit. She had accepted an invitation from the Central Spiritists to speak before their congregation that morning in their six-hundred-year-old church in the Old City. She figured that it wouldn't hurt to shore up her support among the Spiritists and, at the same time, remind the Dexta delegates of her special status in the largest religious organization in the Empire.
She had recited a few passages from the Book of the Spirit, then elaborated on her particular province, Joy. Gloria made it clear that she was in favor of sex, as everyone should be. "I have been reminded lately," she said, "that not everyone agrees with the wisdom of the Spirit, and that many of our fellow beings embrace other systems of belief. The other day, a woman told me that she thought I was a disgrace and that my behavior somehow degraded women." The parishioners hooted and jeered at that, but Gloria quickly quieted them.
"As you may expect, I disagreed with her. But I believe that she did have a point, and it is one that I have pondered, and that all Spiritists would do well to keep in mind. We are a diverse Empire, with many species, many beliefs, many points of view. Even though we Spiritists constitute some seventy percent of the population, we cannot simply ignore the deeply held beliefs and attitudes of the other thirty percent. Just as we may consider many of _their_ beliefs to be silly or oppressive or simply wrong, many of them find _our_ beliefs and behavior to be profoundly offensive. What are we to do about this?
"Well, we could exterminate them. Wipe out the infidel minority who disagree with us. Throughout the course of history, many groups have tried to do just that, when confronted with nonbelievers. But the Spirit counseled not only Joy, but Peace, so that option is closed to us, as it obviously should be.
"Nor can we forcibly convert those who disagree, for the Spirit also taught us Compassion, Tolerance, and Generosity. So I believe that the answer must lie in the remaining two Seeds of Wisdom—Love and Knowledge. Long before the Spirit's Visitations, other religious leaders told us that we must love our fellow beings—even our enemies—and the Spirit reminded us that we must know them. In other words, we must learn to love them. How are we to do that?"
Gloria had paused at that point and grinned at the vast congregation. "Beats the hell out of me," she said.
"My point, of course, is that we all have much to learn—about life and love and our fellow beings. Even Avatars of the Spirit don't have all the answers. _I_ certainly don't. But in my role as a Dexta official, I have tried to learn as much as I could about the many ways of life in our Empire. Indeed, it was that desire to learn about our fellow citizens that first drew me into Dexta. Now, as I exercise my various powers and responsibilities within Dexta, I try—always—to keep in mind the fact that my knowledge and wisdom are limited, and that I can only strive to be wiser tomorrow than I am today. I believe that is the most—and the least—that any of us can do. It is what the Spirit asks of us, and what our humanity requires. Thank you for inviting me here today, and may you keep the Spirit with you always."
As Arkady Volkonski escorted her out of the church, she shook her head and said, "God, the bullshit that comes out of my mouth sometimes!"
"It wasn't bullshit, Gloria," Volkonski replied.
"Wasn't it?"
"I didn't think so."
"Really?"
"Really."
Gloria smiled in bemusement. "Maybe there's hope for us all, then," she said.
JILL NOTED GLORIA'S ARRIVAL. LIKE MOST OF the people on the beach, including Jill, Gloria was nude, but she was flanked by Arkady Volkonski and three of his Bugs, all of whom were clothed and looking uncomfortable about it. Of course, they needed clothing to conceal their weapons, for Bugs were _never_ off duty.
Jill squirmed and sighed as Eli applied some more suntan lotion to her breasts and belly. Pheromones aside, the man had an almost instinctive feel for what a woman wanted, and how she wanted it. Sex with him had proved to be an astonishing experience for her; she had reached physical and emotional highs that were beyond anything she had known. She couldn't help wondering if sex was like that for Gloria all the time. It would explain a lot.
At the same time, a quiet voice of caution somewhere deep inside her was never still. Jill was well aware that a man like Eli Opatnu would never limit himself to a single woman. There was no reason he should, when he could have literally any woman he wanted. She was grateful that—for the moment, at least—he wanted her, but she knew it wouldn't last. Eli was based in Manhattan, so there would undoubtedly be other opportunities even after they returned to Earth, but there was no prospect for the kind of relationship that she had always wanted. Her marriage hadn't worked, and she freely admitted that it had been mostly her fault; but that didn't mean she had given up on finding a lasting and devoted partnership. But Eli wasn't the man for that.
In love, as in her life, Jill resisted compromise. As Eli had pointed out, that was not a realistic attitude, but it seemed to be the one she was stuck with.
Gloria greeted them, then ran across the beach and plunged into the surf. Jill watched her, then watched Eli watching her. "What's it like for you and Gloria?" she asked him.
Eli looked at her with a quizzical smile. "Comparisons are invidious," he said.
"No, I didn't mean that. I just wondered what it was like for you genetically enhanced types. Is it just more of the same, or is it something different?"
Eli shrugged. "I have no basis to make such a comparison, since I don't know what it's like not to have my enhancements. It's all subjective, you know, for everyone. When I make love with you, it's a sublime experience. When I make love with Gloria, it's a sublime experience."
"But is it sublimer?"
" 'Sublimer'?"
Jill punched him in the ribs. "You know what I mean. I'm going to start calling you Eli the Eel because you're so damn slippery. You wiggle out of everything."
"A useful survival trait," he said. "If eels weren't slippery, there wouldn't be any eels."
"So you won't talk, huh? Well, maybe I can _make_ you talk." Jill grabbed him and pulled him down to her.
"I DON'T LIKE THAT GUY," PUG SAID TO PETRA on another blanket not far away.
"Why?" Petra asked. "Is it because he gives you an inferiority complex?"
Pug reflexively looked down at himself, then quickly looked back at Petra, who couldn't help smirking. She leaned over and kissed him on the cheek. "Don't worry about it, Cowboy, it ain't the size o' the shootin' iron that counts, it's the aim."
"I just don't like the way he thinks he can always get anything—or any _one_ —he wants," Pug said.
"I feel that way about Gloria, sometimes," Petra admitted. "I don't think normal standards work for people like that. I think you just have to accept them as they are."
"Maybe," Pug said. "But I still don't like him."
"Oh, stop being so...so _male._ You just resent him because he's invaded your little hunter-gatherer group and upset what you thought of as a set of stable relationships."
"What?"
"You heard me. And you needn't act so dumb—you went to college, same as I did. OSI is like a little family group. Gloria's the mother, Jill's your big sister, and I'm your mate. So Eli shows up and scores with Mom and Sis, and now you're afraid that he'll bag me, too. Don't worry, he's not my type." Petra took a closer look at what was going on next door and added, " _Although_..."
"Very funny," Pug sniffed. "Anyway, Eli would have to wait in line behind Whit, wouldn't he?"
Petra looked away from him and concentrated on picking some sand out of her belly button. "I thought we weren't going to talk about that," she said.
"What should we talk about, then? Maybe we could talk about the way you totally ignored my wishes and went ahead and downloaded all of that Bartholemew data into the Dexta computer? Or how about the way you carried on at the opening night reception?"
"Carried on? What do you mean, carried on?"
"You know what I mean. Practically naked like that, in front of everybody."
"Well, _you_ bought me that pareu, you know. You're the one who keeps telling me I should be more like Gloria."
"Well, I didn't mean you should do it at a place like that. You embarrassed me in front of my family."
"And how do you think I felt, seeing you with your arm around Steffany?"
"You were so drunk, I'm surprised you noticed."
"Fuck you, rich boy!"
"WHAT IS WRONG WITH THIS PICTURE?" GLORIA said to Althea as they stood together in the surf. Strategically scattered around the OSI group, Volkonski and his Bugs kept a stoic watch. To their right, Elaine was industriously screwing a man she had just picked up. Directly in front of them, Jill and Eli were writhing like two devout Spiritists in Central Park on a Visitation Day. To their left, Pug was angrily stomping away from the blanket where Petra lay with her back to him.
"Trouble in Paradise?" Althea wondered.
"Could be," Gloria said. "I think we're going to lose Pug. Maybe Petra, too, although from the look of things at the moment, I'd say probably not."
"And what about you? Are we going to lose you, too, Gloria?"
Gloria sighed. "I honestly don't know, Althea. Maybe I should just flip a coin and get it over with, one way or the other."
"You'd make a marvelous Empress, Gloria, you truly would. But I'd hate to lose you."
Gloria smiled at her. "Don't get carried away, Althea," she said.
"I know we've had our differences in the past," Althea said, "but I don't think I've ever told you how grateful I am to you. There I was, stalled at Thirteen for years and years, and then you plucked me out of my little cul-de-sac and brought me into OSI as a Twelve and gave me all these wonderful assignments. Did I ever tell you how much I _adored_ running the Emporium on Sylvania?"
"I thought you might."
"And it's been ever so much fun arranging for the reception. Oh, did I tell you? I found a band that plays those little twentieth-century blues numbers you like so much. I don't understand it, myself—all that caterwauling and wailing about lost dogs and lost women and such—but I knew you'd love it, so I signed them up as soon as I heard them."
"That's great, Althea. Thanks."
"The very least I could do."
They turned and watched the scene on the beach for a few moments.
"Do you think I could seduce Arkady?" Althea asked.
"Althea"—Gloria chuckled—"I think you could seduce anything with warm blood."
"That's why I asked about Arkady. I'm not sure his blood _is_ warm. I'm not even sure he's got any."
Gloria thought about what Volkonski had said to her coming out of church that morning and said, "He does."
"Oh?" Althea asked in surprise.
"It's not what you think, Althea. But I learned something about him today. He's deeper than I thought he was. And you know what?" Gloria grinned self-consciously. "So am I."
NORMAN MINGUS ARRIVED ON NEW CAMBRIDGE the next day. He occupied an entire floor at the Imperial Cantabragian, and late that evening, following a round of receptions, Gloria went up to see him. She made her way through a phalanx of security and a half dozen aides and finally found him in his bedroom, clad in pajamas, a robe, and slippers, sitting in a comfortable chair with a glass of warm milk and some cookies on a table next to him. Gloria sat down in a chair next to his.
"Are those chocolate chip?"
Mingus looked up from some papers he had been reading and noticed Gloria for the first time. He took in the sight of her, resplendent in a low-cut black micro-dress. He smiled, then handed her one of his cookies. "For an Avatar of Joy," he said, "I believe I have a few chips to spare. But don't get greedy."
Gloria munched the cookie. "Just one," she assured him. "I'm getting fat." She slapped her mostly bare left hip.
"Not from where I'm sitting."
"Mind if I take off my shoes?" Without waiting for a response, Gloria slipped off her high heels and started massaging her aching feet. "I tell you, Norman, late nights and high gravity can be hard on a girl."
"I hear you've been keeping very busy."
"Maintaining a high profile, showing the flag for OSI."
"Quite effectively, I gather. What's all this I hear about a 'Fifth Quadrant'?"
"Oh, it's just a sales pitch. The idea is that OSI will serve as a nucleus for all the Dexta offices that aren't effectively represented by the Sectors or Quadrants."
Mingus nodded thoughtfully. "It's not such a bad idea," he said. "Although, from my point of view, four Quadrants are more than enough. But it's a valid point you make, and it strikes me as a useful ploy in your squabble with the Quad Admins."
"It's not just some little squabble, Norman," Gloria said. "They mean to eviscerate the OSI."
"Well, you'll just have to keep them from doing that, won't you?"
"I may need your help."
Mingus shook his head. "I can't openly oppose the Quad Admins, Gloria. And you know why. I'll referee and keep the fight fair, if I can, but don't expect me to intervene on your behalf. In any event, you seem to be doing pretty well without me."
"You think so?"
"I'm not as isolated up there in my ivory tower as you think. My sources tell me you've won a lot of support here this week. If the Quad Admins try to slap you down too blatantly, they'll only be making trouble for themselves. Grigsby and Chandra certainly understand that, and Algeciras will keep his finger to the wind, as always."
Gloria looked at him. "And DuBray?"
Mingus leaned back in his chair and crossed his right leg over his left knee. "I've known Cornell a long time," he said. "I don't always approve of his methods, but I can't argue with the results he gets. He's the most effective administrator I've ever known. He can be remarkably pigheaded, but he's not a fool. You can't beat him, but if he can be made to see that defeating you is not necessarily in his best interests, you may be able to live with him. More importantly, he may decide that he can live with you."
"Maybe I _can_ beat him," Gloria said. "Norman? I think DuBray was involved in the diversion of that Savoy shipment. I don't have the proof I need yet, but—"
Mingus abruptly uncrossed his legs, leaned forward, and took Gloria's wrist in his hand. His watery eyes locked with hers.
"Gloria," he said evenly, "whatever Cornell DuBray may or may not have done fifty-five years ago is of absolutely no moment. I will not have you dredging up the dead past just to gain an advantage in your little turf war. Do you understand me?"
"It may not be so dead," Gloria protested. "We're tracking down that missing Savoy shipment, and I really think we're going to find it. And when we do, I think we'll see Cornell DuBray's fingerprints all over it. He was mixed up in it with Whitney Bartholemew and the zamitat, and they couldn't have gotten their hands on that shipment without DuBray's active cooperation. DuBray's rich, isn't he? Well, where did his money come from? I think if we start digging into his finances—"
"You'll do no such thing!" Mingus released his hold on Gloria's wrist, glared at her for a moment, then collected himself, took a sip of milk, and bit into a cookie. Gloria waited for him to say something, shocked that he had raised his voice with her.
Mingus swallowed the cookie and took another sip of milk. Finally, he said, "Fifty-five years. I can't expect you to understand what that means. It was another time, another age. The things we did then seemed necessary, and so we did them without worrying about what history might say. If Cornell DuBray had anything to do with diverting that Savoy shipment, you can be sure that he had good reasons for it, and lining his pockets was not one of them. You will not investigate his finances, or anything else that does not directly contribute to locating those missing arms. Is that understood?"
"But, Norman—"
"Is that understood?" Mingus repeated more emphatically.
Gloria looked at him in silence for a few moments and saw an intensity in his eyes that she had never seen before. At last, she said, "Yessir. I understand." After a pause, she added, "I mean, I really think I do understand. I met Saffron. I know she was going to marry DuBray, and then something happened and she wound up marrying Bartholemew. I never meant to dig around in your personal affairs, Norman."
"But you must wonder about them."
"That's none of my business. It just happened to come up in the course of the investigation. When I realized there was a link between DuBray and Bartholemew and the zamitat, I automatically figured it was something I could use against him. It didn't occur to me that I might be hurting you in the process. It should have, but it didn't. I'm sorry, Norman."
Mingus was silent for a long time. Gloria could hear the slight rasp in his breathing.
"Gloria," he said, "if I knew anything at all that might help you find those missing arms, I would tell you. Precisely how and why they went astray is another matter. Leave it be. As for hurting me, I doubt that there is anything you could do that would add or subtract a single atom from the pain I have felt for fifty-five years."
"I understand," Gloria said softly. "Norman? Are you going to see her?"
Mingus thought about it for a few moments, then said, "Perhaps."
"I think you should."
"It must be nice to be so sure you know what other people should do."
"I'm sorry, I didn't mean to presume."
"That's quite all right. Ignorant presumption is one of the privileges of youth. Enjoy it while you can." Mingus smiled at her. "Here," he said, "have another cookie."
GLORIA WAS FINISHING HER COFFEE AND A light breakfast with Elaine the next morning when Petra and Pug arrived at her suite. Gloria offered them coffee, which they declined, then rose and joined them in the living room area.
"I asked you to stop by this morning because I met with Mingus last night. By his order, we will no longer be investigating Cornell DuBray and his links to the Savoy shipment. Our one and only objective is to find where those weapons are now. If you were doing any digging about DuBray that is not directly relevant to that goal, stop doing it. Is that perfectly clear?"
"No problem," Petra said. "I've been working on another angle anyway."
"Pug?" Gloria looked at him. Pug shuffled his feet and cleared his throat.
"Uh, Gloria?" he said. "There's something I have to tell you." He looked at Petra and said, "I'm sorry, Petra, I meant to tell you first, but as long as we're here, I think I should just get it over with. I've made a decision, and I'm going to accept that job offer on Pelham."
Petra stared at him and said nothing, but it was clear from her face that Pug's announcement did not come as a complete surprise to her. Pug looked away from her, pulled his pad from a pocket, and tapped a key on it.
"I've just transmitted my formal transfer request to your pad, Gloria. This wasn't an easy decision for me, and I want you to know that I'll always be grateful to you for bringing me into OSI in the first place. It's been a wonderful experience, but this Pelham job is an opportunity I can't ignore."
"I understand, Pug," Gloria said. "I'll approve the transfer immediately. And I wish you all the best in your new position."
Both Gloria and Pug turned to look at Petra.
"Petra?" Pug said after an awkward pause. "That offer of an Undersec position on Pelham is still open. Uncle Benedict says it should take three or four months to free up the spot, and if you still want it then, you can come to Pelham. But...there's something else you should know. Steffany is going to be coming to Pelham."
Petra's eyes widened, but before she could say anything, Pug raced ahead. "Uncle Benedict invited her for a visit. You see, he's _her_ great-uncle, too. Benedict is my mother's father's brother, but he's also married to Steffany's father's mother's sister, so..."
"So you expect me to wait around for three months on Earth while you and Steffany have a high old time on Pelham?"
"Well, I didn't..."
"That tears it!" Petra's eyes flashed with righteous anger. "Gloria, can I bunk in here with you?"
"No problem," Gloria said. "I'll have them put another bed in Elaine's room."
"Great. As soon as I pack up my things..."
"There's no need for you to do that," Pug said. "I'll have the staff collect your things and deliver them here. Petra, I'm sorry I didn't tell you this earlier, but I only came to the decision last night and I—"
"Oh, spare me! Just get out of my sight!"
Pug looked around uncertainly. "Uh, Gloria, I can continue working for you as long as you're here on New Cambridge. I mean, I never intended to leave you shorthanded."
"Under the circumstances," Gloria said, "I think it's best if we terminate your connection with OSI immediately."
"I understand. Well...uh...look, this doesn't affect the OSI reception at my parents' house, of course, so I'll be seeing you there. We can...well, we can say our farewells then." Pug looked once more at Petra, then turned and quickly left the room.
Gloria said, "I'm sorry, Petra."
"Well, I'm not!" Petra insisted. She looked at Gloria and gave a wan smile. "I don't think I would have been very happy in the Ellison family. Too damn many mother's father's uncle's sisters."
GLORIA FOUND CORNELL DUBRAY AT THE Convention Center later that morning. He was surrounded by his usual retinue of security guards and aides, but they cleared a path for her as she approached. "Could I have a word with you in private?" she said.
DuBray nodded, and they soon found themselves alone in an empty conference room. "I spoke with Norman last night," she told him. "He ordered me to discontinue any investigations into your links to the Savoy shipment. Of course, I'll respect his wishes, but I do have one question for you, and I expect an honest answer. Do you know anything—anything at all—about where those weapons went or where they are now?"
The Quadrant Administrator stared at her for a moment, then offered a bemused smile. "You must have a remarkably low opinion of me, VanDeen."
Gloria didn't say anything to that.
"You see me as a villain," DuBray said. "Understandably so, I suppose. So you assume that I must therefore be villainous in all things, at all times. I'm the Evil Cornell DuBray, so I must be capable of any imaginable form of treachery, treason, or betrayal. I might even know where the missing arms are and remain silent merely to protect myself. PAIN terrorists run riot and slaughter people who work for me, yet I just smile and twirl my mustache like the villainous character in a melodrama that you imagine me to be. Really, that's quite insulting."
"So you're saying that you don't know anything?"
"What I'm saying, VanDeen, is that I was serving Dexta and the Empire long before you were born. Who are you to question my loyalty?"
"It's a question that must be raised," Gloria said. "I know you were linked to that shipment, somehow. Bartholemew couldn't have gotten it without your help. You must know something about it."
DuBray turned his back on her, walked a few paces away, then turned around. "You say you spoke to Norman about this? And what did he tell you?"
"Not much. Only that I should let the past remain buried. But those weapons—"
"Are Spirit knows where! If I knew where they were, don't you think I'd have done something about it by now? Or would that be inconsistent with your image of me as an archvillain?"
Gloria thought about it in silence for several moments and realized that DuBray had a point. She had been so eager to bring him down that she hadn't considered the possibility that he might be innocent—or, at least, not entirely guilty.
"All right," Gloria said, "I may have made some unwarranted assumptions about you. But it's still possible that you could know something that would help us find the weapons, maybe without realizing it. Didn't Bartholemew say anything about what he was going to do with them?"
DuBray chuckled mirthlessly. "You don't understand the situation."
"Then help me to understand it."
"If Norman didn't explain it to you, I certainly won't. All you need to know, VanDeen, is that from my point of view, those weapons vanished as completely as if they had fallen into a black hole. When they started turning up again, I was more surprised than anyone."
"But they never went to Savoy, did they?"
"Apparently not."
"They desperately needed those weapons on Savoy," Gloria said. "They might have made all the difference. I've been doing some reading, and—"
_"Reading?"_ DuBray bellowed in evident contempt. "You've read a book or two, and now you think you're an authority on what happened half a century ago? The ignorant arrogance of youth never ceases to amaze me."
"Are the books wrong?"
"The books are... _books_! Spirit, VanDeen, _I was there_! You and the authors of those books were _not._ You cannot possibly know or understand what happened then. Don't speak to me about Savoy. You haven't the right."
"But I do have a responsibility to find those weapons. They took a shot at _me_ with one of them, you know. That gives me the right to ask you about them."
"Perhaps it does," DuBray conceded. "Investigate to your heart's content. Strategically intervene till the cows come home. But don't waste your time asking about Savoy. It simply isn't relevant to your inquiry."
Gloria nodded. "I'll accept your word on that, then."
She started to turn to go, but DuBray stopped her with a look. "I know it may be hard for you to imagine, but we really are on the same side. I want those weapons found even more than you do. Those bastards have been using them in _my_ Quadrant, and they will undoubtedly use them again unless they are stopped. Internal Security is already looking high and low for them, and perhaps your historical inquiries will bear some fruit. I welcome any contribution you can make to the investigation, and I will help you in any way I can. If you have any _relevant_ questions you need to ask me, don't hesitate."
Gloria nodded. "Thank you, Administrator DuBray," she said.
DuBray returned the nod. "You are very welcome, Ms. VanDeen."
THE UTILITY CLOSET SEEMED SPACIOUS NOW. IT was amazing how much difference a 50 percent reduction in the population could make. Petra even had an extra console, which came in handy for the work she was doing.
She had pored over the B & Q records and found that the company had owned no fewer than seven ten-thousand-ton freighters in August of 3163. She meticulously checked the logs and cargo manifests of each of them, but found nothing out of the ordinary. Still, the apparent falsification of the records of the sixty-thousand-ton freighter made her suspicious of anything she saw in the records from that period.
She attempted to be logical and systematic. Fact: The weapons existed and had been shipped to New Cambridge—assuming the _other_ records could be trusted. That was a necessary assumption, because if the shipment had never arrived at New Cambridge, then she was wasting her time anyway. Fact: They had been stored, initially at least, in the orbital warehouse of Stavros & Sons. But the relevant records of the defunct Stavros & Sons no longer existed, so it was impossible to tell precisely how long the shipment had remained at New Cambridge. So she made the reasonable assumption that the weapons had been removed no later than the end of 3163. Fact: The weapons had officially been transferred to the custody of B & Q Shipping for transshipment to Savoy. Fact: Gloria's source said that the weapons had been removed in three shipments.
Therefore: Only B & Q could have removed the weapons from the Stavros warehouse during the specified time frame, and it made sense that they would have used their own ships to do it. So one or more of those ten-thousand-ton freighters owned by B & Q must have done the job. One freighter in three trips, or three freighters in one trip, or some combination that added up to three.
But trips to _where_?
They had to have gone somewhere. Bartholemew hadn't simply dumped them in space, because the weapons had shown up again, fifty-five years later. There was a large but finite number of places Bartholemew could have taken them. Using freighters capable of making planetfall, virtually any world in the Empire could have been the destination. But the logs of the seven freighters in question placed a limit on how far away the shipments could have been sent.
Petra made another necessary assumption: that the logs of the freighters could be trusted. Contracts might be falsified, but a vessel's logs were sacrosanct, or should have been. The information contained in them—hours in Yao Space, light-years traveled—needed to be accurate. Ferguson Distortion Generators and fusion reactors had to be inspected and overhauled at regular intervals. Someone might fudge the destinations and dates that appeared in a log, but it would be foolish—perhaps even suicidal—to tamper with time and distance figures. So hidden in those figures lay the answer to where the weapons had gone.
One more assumption was necessary. She couldn't prove it, but Petra operated on the assumption that all three shipments had gone to the same place. It seemed unlikely that Bartholemew would have broken up the weapons cache and stored it on three different worlds. If he had, then her investigative strategy wouldn't work.
But if they had all gone to the same place, and more than one freighter had been used, then a careful examination of the logs of each of the seven freighters could lead to...
_Headaches,_ Petra thought. It all made sense to her, but finding the needles in this particular haystack was going to take a lot of work. If the freighters had gone directly to their secret destination, then immediately returned to New Cambridge, it would have been easy. But she realized that Bartholemew would have covered his tracks. Almost certainly, each freighter would have proceeded from Destination X to one or more additional ports of call before returning to New Cambridge. If they had simply gone to the secret hideaway, then returned, they would have been flying either with empty holds or with cargoes that had originated at Destination X. So they would have gone elsewhere before returning.
Still, the time and distance figures in the logs put limits on how far the freighters could have gone. Petra calculated that the mystery world could have been no more than eighty-five light-years from New Cambridge.
On the second console, Petra called up galactography charts, and was dismayed to discover just how many planets there were in a sphere of space 170 light-years in diameter. There were, it seemed, no less than 146 Empire worlds encompassed by the sphere. But it was worse than that, because that figure included only established, inhabited colonies. The total number of known _planets_ in that volume of space exceeded ten thousand. Conceivably, Bartholemew could have stashed the weapons on some barren rock or methane-shrouded moon.
But, between the ship logs and the galactic atlas, she had a place to start. Slowly but systematically, she fed assumptions into the computer involving journeys from New Cambridge to any two planets—inhabited or otherwise—within the target sphere. The computer quickly kicked out time and distance figures, which she then compared with the numbers in the logs. Any two-legged itineraries that were numerically feasible, she stored in another file for further analysis.
This could take a while, she realized. On the other hand, what else did she have to do with her time?
"AH. MY TAX CROWNS AT WORK."
Petra looked up and saw Whit Bartholemew at the door of her utility room, grinning.
"What are you doing here? How did you get in?"
Bartholemew shrugged. "I'm not unknown in this city. Getting in was easy enough, but finding you once I was inside took some doing. My word, didn't they even give you a real office?"
"OSI is not universally popular," Petra said.
"Figures. The only thing worse than being outside a fascist bureaucracy is being inside one. They eat their young, you know."
"Look, Whit, I have a ton of work to do, and I don't really have the time to sit here and listen to you insult Dexta. Why are you here?"
"I thought I might take you to dinner."
"I've already eaten." Petra glanced at the crusts of the pizza she'd had delivered.
"If you call that eating," said Bartholemew. "Very well, then, what about a drink? I imagine you could use one after the day you've had."
"You heard, huh?"
Bartholemew nodded solemnly. "Exit Palmer Ellison," he said. "I like the lad well enough, but nevertheless, I say good riddance—for both of us. You really are better off without him, you know."
Petra sighed heavily. "Whit, I really don't want to talk about it."
"Nonsense. Of course you do. You've spent all day staring at these consoles, diligently doing your job, and all the while you've been thinking of all the things you wish you had said to Pug. Well, you can't say them to him, but I stand ready to listen in his stead. What kind of friend would I be if I weren't willing to do that? And as your friend, I insist that you come with me for a drink. I'm much bigger than you are, Petra Nash, and if you won't come voluntarily, I'm perfectly capable of carrying you out over my shoulder."
So Petra went out with Bartholemew. He waved away his limo and they strolled along the busy sidewalks of nighttime Central. As they approached the Dexta complex known as Gibraltar, they were herded across the street by harried police and Security personnel.
"I imagine everyone in Central will be happy to see the Quadrant Meeting end," Petra observed.
"On the contrary. They'll miss all the money your people have been spending. People will put up with almost any amount of official harassment as long as there's something in it for them. In any case, blind obeisance to arbitrary authority is the lot of most of humanity. You'll find more independence and freedom in the average herd of sheep than you will on a city street."
They came to a sidewalk café that looked good to Petra, but Bartholemew had another destination in mind. He pointed upward toward a lofty skyscraper. "There's a restaurant up there with a spectacular view of the city."
"I'm not dressed for a place like that," Petra protested. She was wearing jeans and a clinging white pullover top.
"That won't be a problem," Bartholemew said, "since I happen to own the restaurant. You know, this is the first time I've seen you wearing actual clothes. Very sexy. Makes me want to rip them off."
"Not tonight, Whit."
"The night is still young. At least give me a chance to ply you with soft words and strong drink."
The view from the fortieth-floor restaurant proved to be as spectacular as Bartholemew had promised. From their table, they could see the dark mass of Gibraltar a few blocks away, the variegated architecture of the nearby Old City, and the brightly illuminated vessels plying the harbor. "Central is quite a city," Petra said as she sipped her white wine.
"So were Sodom and Gomorrah," Bartholemew said. "And you know what God did to them."
"Is there anything you don't hate?" Petra wondered.
Bartholemew thought about it for a moment. "I'm rather fond of my mother," he said at last. "And I've taken quite a shine to you. Beyond that..." He shrugged inconclusively.
"Why?"
"Why do I detest that which is detestable? Why do I loathe the loathsome? I was born and reared in the belly of the beast. Is it so surprising that I would grow to hate it?"
"That's too easy, Whit," Petra said. "I was born and reared in a slum, but I don't hate the Empire that created that slum."
"You should. But instead of rising up against your oppressors, you joined them. That's all too common a story. Perhaps if someone had shown you a different path, you might have followed it, instead of the well-worn trail you chose."
"I'm not sure which bothers me more," Petra said, "your cynicism or your hypocrisy. If you really believed half the things you say, you'd have gone off somewhere and joined PAIN."
Bartholemew laughed.
He raised his glass in a toast. "Up the rebels," he said.
Petra shook her head. "I will never understand rich people. Gloria was the only one I knew, and she always seemed pretty well adjusted. Then I met the Ellisons and discovered what ten generations of inbred arrogance can do to people. And now you, with your ridiculous posing and prosing, damning the establishment while you sip expensive wine in your own restaurant."
Bartholemew reached across the table and cupped Petra's cheek in his right hand. "You know what your problem is, Petra Nash? You expect the world to make sense. Trust me, it doesn't."
"Tell me about your father," she said.
"Now, why would you want to know about such an odious subject as that?" Bartholemew leaned back in his chair and drank some more wine.
"I've been trying to get into his head," Petra told him. "Didn't he ever say anything to you about that Savoy shipment?"
"The old man and I weren't much for father-to-son heart-to-hearts. When I was a boy, we mainly ignored each other. By the time I was an adolescent, our relationship had blossomed into mutual contempt, which lasted as long as he did. I'm not sure when I first realized that he was a zamie, but somewhere along the line it became clear to me that the Bartholemews were not quite as respectable as my mother wanted everyone to think we were. In any event, he rarely talked about his work."
"What kind of man was he?"
"Self-contained. Short-tempered. Dictatorial."
"I suppose that accounts for your attitude toward authority."
"I suppose it does. A reasonable response to my situation, wouldn't you say? I never had the luxury of—"
Bartholemew stopped short as a dazzling, blue-green flash illuminated the city. It seemed to emanate from beyond Gibraltar, and was followed an instant later by a second brilliant eruption from the Old City. There was a third flash, coming from somewhere off to the left, then three deafening thunderclaps in rapid succession.
Central was under attack.
PETRA HAD NEVER BEEN IN SUCH A HIGH-LEVEL conclave, and felt even smaller than usual. She had never seen so many single-digit Dexta brass gathered in one place. The conference room in Gibraltar was packed with interested parties, of whom she was the lowliest. She sat between Gloria and Arkady Volkonski at the big conference table, trying not to gape.
At the head of the table sat Norman Mingus himself, whom Petra had seen but never met, looking craggy and imperturbable. At the far end sat Cornell DuBray, a study in icy composure. Arrayed between them were Dexta's Internal Security Administrator, Gavin Chang (Level V), the Quadrant 4 IntSec chief, Elizabeth Irons (Level VI), and a flock of Level VII's. An aide to President Edwin Ogburn of the Republic of New Cambridge was in attendance. Also present was General Nelson Alvarez, deputy chief of Imperial Security, who was not in Dexta but effectively outranked everyone there except Mingus.
One of the Sevens was reciting a summary of the damage from last night's attack. "The Imperial Museum in the Old City sustained major damage, including the collapse of the roof of the East Wing. Damage to the collections has not yet been determined, but it is likely to be considerable. The New Cambridge Department of Revenue Building is gutted, a total loss. And Dexta's Old Annex is about seventy percent destroyed, with heavy damage to the parts of the structure that are still standing. Casualties, as of ten o'clock this morning, are put at 173 dead, 543 injured, and at least 100 missing."
_And one of them could have been me,_ Petra thought. If Whit hadn't lured her away when he did, she would have been in the Old Annex when the terrorists' bombs destroyed it. The thought somehow resisted analysis. Maybe it was one of those things that you couldn't think about very much without going crazy. Petra suppressed a shudder and tried to concentrate on the meeting.
No one said anything for a moment after the Seven had finished. Then Elizabeth Irons cleared her throat. "It's still very early in our investigation," she said, "but the three detonations seem to have been of approximately equal force, so presumably similar devices were used in each attack. My technical people say that the energy expended was equivalent to what you would get if you simultaneously detonated a cluster of perhaps six or seven plasma grenades. Thanks to the OSI report"—Irons glanced quickly at Gloria and Petra—"we know that such grenades were included in that missing Savoy shipment. So it seems likely that they were used in the attack."
Mingus looked at the presidential aide. He straightened up in his chair and said, "As you know, President Ogburn has declared a state of emergency, planetwide, and called up several militia units. We were going to declare martial law in Central, but I gather that would be superseded by the Imperial declaration."
"Correct," said General Alvarez. "The city of Central and everything within a hundred kilometers of it has been under a decree of Emergency Imperial Rule since six this morning. All Imperial Marine units on the planet are on full alert, and the Navy has increased orbital patrols to the limit of their capability. We've sent out couriers asking for additional ships from other commands, but it will be several days before any of them get here."
"Thank you, General," said Mingus. He turned to look at Gavin Chang, who was known around Dexta as the Boss Bug. "Gavin, I understand you have some late news."
"Yes, Mr. Secretary. Just before we convened this morning, I was informed that PAIN has released a communiqué claiming credit for the attacks. The usual ideological claptrap, but they promise more and bigger attacks. It's significant that they struck Imperial, Dexta, and planetary targets, meaning that we will have to increase security for all three categories. I would point out, Mr. Secretary, that last night, they hit only soft targets. That implies a limited capability."
"Or maybe they're just building up to the big one," suggested Elizabeth Irons. "We have to assume that they have the quadrijoule plasma bombs from that Savoy shipment."
"If they do," asked the presidential aide, "why didn't they use one last night? They could have destroyed the whole city at one time instead of doing it piecemeal."
"Maybe they're waiting for a bigger target," said General Alvarez. "The Emperor is due here at the end of the week. I've sent a courier recommending the cancellation of his trip, but, of course, the Household would never agree to that. The Emperor can't be seen to be intimidated by terrorist threats, so they'll go ahead with the trip no matter how risky it is. If PAIN sets off a big one while he's here, they could effectively decapitate the Empire. Secretary Mingus, if we can't keep the Emperor out, maybe I can persuade you to leave."
Mingus smiled grimly. "Not possible," he said. "I have to stay, for the same reason that Charles cannot stay away. I assume that Imperial is taking all necessary measures to assure the Emperor's safety while he's here?"
"We're doing everything we can," said Alvarez. "We got one break, though, in that the Emperor won't be staying in or near the city. He'll be staying at the estate of Lord Brockinbrough, about sixty kilometers north of Central. So even if the city goes up, he should be safe there."
"Well, what about the people of this city?" asked the presidential aide. "I'd like to recommend to President Ogburn that we begin an immediate evacuation of nonessential personnel. There are a hundred million people within fifty kilometers of where we sit, and every one of them would be killed by a two-hundred-quadrijoule bomb."
"Where are you planning to send them, Bill?" DuBray asked dryly.
"Well..."
"That would never do," Mingus said, shaking his head. "If PAIN ever got the notion that they could force mass evacuations of major cities just by setting off a few grenades, we'd never see the end of it. I realize that for political reasons, President Ogburn must be seen to be doing something about all of this, but we can't countenance an evacuation. Certainly not at this point."
"Our best bet," said Chang, "is to locate PAIN's base of operations and put the kibosh on the bastards before they can spring their next move. If they have those plasma bombs, they've got to be storing them somewhere."
"Agreed," said Mingus. "And in that connection, I believe OSI has something relevant. Ms. VanDeen?"
"Yessir," Gloria said. "My assistant, Petra Nash, has been investigating what happened to that Savoy shipment in 3163. Petra?"
Petra hadn't realized that she would be required to say anything. She swallowed hard and cleared her throat a couple of times. "Uh...well...I think I may be able to trace what became of that arms shipment after it was removed from New Cambridge. We have some old records that should allow us to narrow down the possibilities, at least. The thing is, I was working in the Old Annex, and, well..."
"Petra still has the original data she was using, stored in her pad," Gloria said. "Ms. Irons, if you could get her some office space here in Gibraltar, she could continue her investigation."
Irons nodded. "I'll see to it," she said. "Ms. Nash, you can resume your work immediately after we're through here."
"Thank you, ma'am," Petra said politely.
"That's all very interesting," said Alvarez, "but I don't see what it does for our present situation. Just because the weapons were stashed somewhere fifty-five years ago doesn't mean that they're still in the same place."
"Maybe not," Gloria agreed, "but finding their initial destination may give us a lead to where they are now."
"We'll let OSI concentrate on that possibility," said Mingus. "In the meantime, Internal Security and Imperial will continue their ongoing investigation. Gavin, you mentioned that they hit soft targets last night. I assume we are going to harden any other such targets."
"To the extent that we can," said Chang. "But IntSec is already stretched thin. Local law enforcement is also fully committed. The unpleasant fact is that there are a lot of potential targets that are only minimally protected."
Petra knew what Chang meant. She thought of how easy it had been for Whit to gain access to the Old Annex last night. Not to mention the pizza delivery guy.
"And if they have a big bomb," Irons added, "they could put it anywhere and still destroy the city. I'd like to do a full sweep of Central and the surrounding region, but we just don't have the personnel to do something like that in a reasonable amount of time."
"We can do a random sweep," said Chang, "and maybe we'll get lucky."
Mingus frowned at his Security chief. "It always distresses me," he said, "when I hear my Security people use the word 'luck.' " He continued staring at Chang for another moment, then said, "All right, you know what needs to be done, so let's all get busy doing it. We'll meet again tomorrow morning."
Mingus got to his feet, then everyone else did. Gloria turned to Petra and said, "I'm going to give Jill a call and tell her to get over here and help you. Schmoozing will have to wait."
FASTER THAN SHE WOULD HAVE BELIEVED POSSIBLE, Petra had her own office in the Internal Security section of Gibraltar. Whatever prejudice the locals might have had against OSI had been put in abeyance for the duration of the crisis. The Bugs helped her get settled and even brought her coffee.
All the work she had done yesterday was lost when the Old Annex was destroyed, but Petra still had the B & Q files in her pad. She downloaded the data into her console in the new office and got to work. It occurred to her that she had been doing things the long way around yesterday, and came up with a more streamlined search strategy. She set things up the way she wanted and had just begun work when Jill arrived.
Jill sat down next to her and stared at the console screen while Petra explained what she was doing. "These are the seven freighters I'm interested in," Petra said, indicating a list on the console screen. "In the last four months of 3163, they made a total of thirty-one trips. My assumption is that three of those trips involved hauling the arms shipment to what I'm calling Destination X. There are logs and cargo manifests for each freighter and each trip, but they can't be trusted."
"That's for sure," Jill agreed. "One thing I've learned in looking into this double-flagging business is how easy it is for documents like that to be altered. And that would be business as usual for someone like Bartholemew if he was in the zamitat."
"Nevertheless," Petra said, "I'm assuming that the records for total light-years and Yao Space hours for each vessel are accurate. I'm also making the assumption that the records for the final, homeward leg of each trip are correct. If Bartholemew was using these trips to transport the arms, he wouldn't want any questions raised about the voyage when the freighters returned to New Cambridge."
"Makes sense," said Jill.
"Okay, then, here's what I'm doing with each trip for each freighter. We'll start with this one, for Freighter Number One. It departed New Cambridge on August 30, returned on September 19, having traveled a total of 141 light-years. The final leg, according to the records, was 52 light-years, from Halcyon to New Cambridge. The logs list two previous ports, but we'll ignore those. What we have to look at is the other 89 light-years of that journey. I'm assuming it made at least one stop—at Destination X—and maybe one other stop to pick up cargo to take to Halcyon. The computer is going to look for any combination of either one or two ports or planetary systems that would add up to 89 light-years, beginning on New Cambridge and ending at Halcyon."
Petra tapped a key and the computer processed the problem and displayed its results a second later.
"Yikes!" Petra exclaimed. The computer had kicked out forty-seven possible routes for Freighter Number One. "I didn't think there would be that many."
"It's a bunch," Jill agreed. "Now what?"
"Now we go through the same routine for each of the other thirty trips. Then we'll compare the results and look for three trips to the same place. But I'm afraid we'll get a hell of a lot more than three. Might as well get started. I'll take Freighters One through Four, you take Five through Seven."
"Gotcha," said Jill.
GLORIA STOOD IN A QUIET CORNER OF THE South Central High School gymnasium, dictating notes into her pin-pad. The reception for Sector 19, Division Gamma-Five, was a sparsely attended, low-key gathering, with second-rate eats and a band that played a dismal selection of popular songs from the thirtieth century. Ahhh, the glamour!
She had met an assortment of lower-level Dexta folk, resolutely giving each of them her standard OSI-is-wonderful spiel. She had danced with many of the men, posed for scores of handshake-and-a-smile pictures, and generally provided a touch of sophistication and glitter to what was, by any standard, a tedious affair. But the job had to be done, even if the Empire was going up in flames all around her.
People were jittery tonight, and the forced gaiety was a strain. There was solace only in drinking—or so it seemed from the heavier-than-usual consumption of alcohol—and the knowledge that the very obscurity of the reception probably provided them with an additional measure of security. PAIN wouldn't waste its weapons on a two-crown affair like this one.
Gloria wasn't so sure of that. This certainly qualified as a "soft" target, and another well-placed cluster of grenades could wipe out a couple of hundred Dexta people—including, not incidentally, the most famous of all. Volkonski and three of his OSI Bugs were there with her, in addition to the normal Dexta and local security, but Gloria was acutely conscious of her special status as a high-priority target. The smell of burned hair was something she was not likely to forget. And you never knew when another Eloise Howell might turn up.
Still, there was some odd comfort in the knowledge that PAIN wasn't simply trying to kill _her._ They were trying to kill _everybody._ And if they had one of those plasma bombs secreted somewhere in the city, they might very well succeed.
But they would wait, she knew. They wouldn't detonate the big one until Charles was in the city.
She hadn't realized that the Brockinbroughs had an estate on New Cambridge. That no doubt meant that Cousin Larry would be accompanying Charles on the trip—an unappealing prospect for Gloria. She had hoped to find some time alone with Charles, but she could hardly avoid Larry if Chuckles was going to be rooming with him.
She had been far too busy to give much thought to the whole question of becoming Empress. It was just too big to deal with. She would prefer to continue delaying and deferring it, but Charles would not wait forever. He needed an heir, so he needed an answer, and Gloria could not avoid making a decision. Just what that decision would be eluded her completely. The appeal of becoming Empress was undeniable, but things seemed to be looking up for OSI. At the meeting this morning, OSI had a place at the table right alongside all the high Dexta and Imperial muckety-mucks, and even Cornell DuBray had not objected to her presence. And most of the people she had met here this evening seemed to be in favor of OSI.
"Durward Inglesby," she said into her pin-pad, "Sector 19, Assistant Deputy Sector Admin, Level Eleven. Happily married, he says, with three kids. Mentioned that two of the people in his office have applied for transfer to OSI. Got the impression that he wouldn't mind losing one of them, but didn't say which one. Check into it."
"And what about me?" Eli Opatnu said over her shoulder. "Would you accept _my_ application for transfer into OSI?"
"In a heartbeat," Gloria said, spinning around to face him. "Unfortunately, we aren't rated for a Seven. You'd have to accept a demotion to say, Thirteen. That would involve making coffee and picking up my dry cleaning. Think you're up to it?"
"For you, anything!" Opatnu gave her a kiss that singed her lips.
Gloria took a deep breath, sucking in pheromones by the billions. She'd been too busy even for sex, and hadn't been with anyone since that night with Eli and the others in the null-room. But that old urge to merge had been building up within her, and Opatnu, in his natty Imperials, was looking better than ever. On the other hand, there were potential complications.
"How are you and Jill getting along?" Gloria asked him.
"Lovely lady, Jill," Opatnu said. "Not the equal of her even lovelier boss, of course." Opatnu stepped back a pace to take a better look at Gloria, who was wearing nothing but a silver-and-blue pareu, fastened with a turquoise brooch.
"Her lovely boss," Gloria said, "wouldn't want to get in the way of anything."
"I think Jill understands the situation."
"Nevertheless, I don't want people who work for me to think I'm interfering in their love life."
"Laudable of you. Still, allow me to point out that Jill is not here tonight and doesn't need to know anything about this."
"True," Gloria agreed. "You know, I think I've done my quota of schmoozing for this evening."
"Strangely enough," said Opatnu, "so have I."
GLORIA AND ELI SAT NEXT TO EACH OTHER ON a sofa in her suite, sipping expensive brandy, following an explosive interlude in her bedroom. Sex with Eli, augmented by a little jigli and Forty-eight, was a searing, incandescent experience, like nothing she had ever done before. Even Charles had never matched Eli's volcanic sexuality, although she wondered what Charles might achieve with the aid of Forty-eight. She might have a chance to find out, she discovered, when Eli offered her a present.
He pulled a handful of lozenges from his pocket and carefully deposited them on the coffee table—half a dozen purple pills, and as many green ones.
"Forty-eight," Gloria cried in delight.
"Purple's the Forty-eight," he explained, "and green is the neutralizer. Just in case. Enjoy."
Gloria gave him a kiss, then picked up one of the purple lozenges and examined it. "Back on Earth," she said, "I was told that it wouldn't be ready for full distribution for at least another six months. Do you know if they've been having problems with it?"
"Some, from what I hear. In spite of the warnings, some people get carried away and crunch down on it. There have been some deaths reported. I gather that there's some debate about whether they should routinely give out the neutralizer along with the Forty-eight. That might just encourage people to go ahead and crunch down. Anyway, I expect they'll get it all sorted out before long, and you'll be able to get Forty-eight on Earth and throughout the Empire. In the meantime, just be careful with it."
"I'll think of you every time I use it," Gloria told him.
"I can't imagine a greater compliment," he said. Opatnu pulled her closer and kissed her.
Before they had broken, the outer door of the suite opened, and in walked Petra and Jill. Gloria saw the sudden look of hurt on Jill's face, but was immediately distracted by the look of triumph on Petra's.
"We got it!" Petra declared. "We know where they took the weapons!"
THE SAME HIGH-POWERED GROUP AS THE PREVIOUS morning had assembled in the conference room at Gibraltar, and Gloria watched with a feeling of pride and accomplishment as Petra stood and delivered a summary of her findings. Her assistant looked smart, sexy, and confident in her gray skirt, blue blazer, and half-unbuttoned white silk shirt. After the way things had worked out with Pug Ellison, Gloria was glad to see Petra rebound so quickly. Not many Thirteens ever got the chance to brief Norman Mingus.
Petra stood next to a large display screen and explained her search strategy quickly and efficiently, then turned to describe what was on the screen. "This is a representation of the local region of space, centered on New Cambridge," she said. "After the first stage of our search for possible routes for the thirty-one trips by those seven freighters in the final four months of 3163, this is what we wound up with." The screen suddenly blossomed in a profusion of arcing red lines that looked like the tracks of subatomic particles in the beam of a high-speed collider.
"Three hundred and fourteen possible routes," Petra said. "Then we looked for planetary systems that could have been visited at least three times." Petra clicked a key in her palm pad, and most of the red lines on the screen disappeared. "That brought us down to thirty-seven possible destinations. However, if you look at each individual route, you find that eighteen of them don't make sense, practically speaking. This one, for example, goes out ninety-six light-years, then doubles back on itself fifty-three light-years, then out again twenty-seven light-years. It seemed unlikely that anyone would plot such a path, so we discarded such cases."
The screen responded to Petra's command, and nearly half of the remaining traces vanished. "That left us with nineteen possibilities. But eleven of them plot courses in the direction of Ch'gnth territory. It's highly unlikely that anyone would have gone in that direction in the fall of 3163. Even after the Empire victory at Savoy in September, the Ch'gnth still had plenty of power in the region. It would make no sense to transport the weapons to a place where they might fall into the hands of the enemy. In addition, I doubt that Bartholemew would have risked his ships that way. So now, we were left with just eight possible destinations."
Gloria noticed that the people around the table were beginning to lean forward slightly. Petra had their complete attention.
"Of the remaining eight," Petra continued, "I eliminated Halcyon. It seemed unlikely that they would transport the weapons from one big, busy, well-regulated port to another. And although it's smaller than Halcyon, I eliminated Ifni for the same reason. Two more, I eliminated because they were too small. Parker's Planet has nothing on it except a minor Imperium mining operation. And St. Regis is just a tiny Catholic utopian community, centered around a monastery. Bringing the weapons to either place would have attracted too much attention. So now, we're down to just four.
"Each of the four is an uninhabited planetary system. They don't even have names, just Imperial Survey numbers. This one," Petra said, tapping a spot on the screen, "has no terrestrial planets, just gas giants and some very small moons. And this one is a variable star that flares up every sixteen months. It didn't strike me as a likely destination, so I eliminated that one, as well."
Petra paused, glanced quickly at Gloria, then continued. "Of the remaining two, this one, GAC 4361, has two terrestrial planets. But one of them is a Venus-clone, and the other is smaller than Mars, very cold, and has almost no atmosphere. Which brings us to GAC 4367."
A single red dot remained on the screen. "The third planet from this star was first surveyed by the Terrestrial Union nearly nine hundred years ago. It's a scumworld."
As humans advanced into the galaxy, they had discovered that life was quite common, but usually not very interesting. On most planets where life had evolved, it was similar to the sort of life that had dominated Earth for more than three billion years: pole-to-pole mats of one-celled, blue-green algae. They came to be known as scumworlds.
"As you know," Petra said, "in the early era of human expansion, prime real estate was hard to find, and the old Terrestrial Union embarked on an ambitious program of terraforming. Scumworlds were considered candidates because the algae had created an oxygen-rich atmosphere. Anyway, the Union established a small, preliminary base on GAC 4367-III in 2297. Not much was accomplished, terraforming went out of vogue, and the base was abandoned in 2349. As far as we know, that base is still there. It's just forty-three light-years from New Cambridge, and I think that's where Bartholemew took those weapons."
There was a long moment of silence around the conference table. Finally, Norman Mingus said, "An excellent piece of work, Ms. Nash. Thank you very much." Petra tried but failed to suppress a grin as she returned to her chair. Gloria reached over and gave her hand a squeeze.
Mingus looked around the table. "Well?" he asked. "Any comments?"
General Alvarez scowled and said, "Pretty damn thin, if you ask me."
"You do make a lot of assumptions, Ms. Nash," said Elizabeth Irons.
"How do you know they didn't stash them in the basement of that monastery on St. Regis?" asked Gavin Chang. "Or at the bottom of a mine on Parker's Planet?"
"For that matter," said Alvarez, "I don't see how you can justify eliminating Halcyon or Ifni."
"Well," Petra said, with obvious uncertainty in her voice, "it was necessary to make some assumptions, but I tried to put myself in Bartholemew's head and look at things the way he would have. I just think that GAC 4367 is the most likely possibility."
"We're not claiming certainty," Gloria quickly put in. "Maybe those weapons _are_ hidden in that monastery. But the most reasonable assumptions all lead us to that scumworld Petra found. From Bartholemew's point of view, it would have been perfect. An uninhabited world with an oxygen atmosphere, and storage facilities already available at the old Terrestrial Union base."
"A base that was built over eight hundred years ago," Alvarez objected. "For all we know, it collapsed centuries ago."
"Maybe," Gloria conceded, "but those old terraforming projects were supposed to take a couple of thousand years. They probably would have built that base to last."
"I question the entire basis for this line of investigation," said Chang. "Wherever those weapons went originally, is it reasonable to assume that this Bartholemew character absconded with them simply for the purpose of leaving them on some deserted hunk of rock for half a century? Seems to me, he would have sold them off at some point. Even if Ms. Nash's assumptions are correct, those weapons were probably removed from that scumworld decades ago."
Gloria noticed Mingus and DuBray staring at each other from opposite ends of the table. Mingus nodded fractionally, then looked at Chang. "Your point is well taken, Gavin," he said, "but we can only work with the information now available to us. For our immediate purposes, I believe we should discount any speculation about what might have become of those weapons in later years. I think Ms. Nash has given us, at a minimum, a reasonable place to start. The question becomes, what do we do about it?"
"Isn't that obvious?" Gloria asked. "I think we should send a LASS and a company of Marines to GAC 4367 immediately."
If it was obvious to Gloria, it was anything but to General Alvarez. "Ms. VanDeen," he said, "do you think Land-Air-Sea-Space vehicles grow on trees? Do you think Marines grow like weeds? Our resources here are already stretched to the breaking point. The Navy is now searching every last vessel that enters the New Cambridge system, and it has to be done _before_ they reach port because the Orbital Station itself could be a target. We simply don't have any ships available for fishing expeditions."
Gloria turned to Chang. "Well, what about Dexta? Internal Security has its own ships."
"All of which are engaged in the interdiction operation," Chang responded. "Nor do we have personnel available to be diverted from their present responsibilities."
Gloria could see where this was headed, and she didn't like it. She turned to Petra. "Are there port facilities or landing strips on that planet?"
"According to the old Terrestrial Union charts we found, there's no landing strip, but the base was built on the shore of a big bay. They must have had a dock."
"Good. Then we can take a Cruiser." She looked at Mingus. "Mr. Secretary," she said, "unless you have any objections, on my authority as head of OSI, I intend to take a Cruiser and half a dozen of our Bugs and go check out that planet. Forty-three light-years is only about thirteen or fourteen hours in a Cruiser. We can leave this afternoon and probably be back by tomorrow evening."
"And just what would you expect to accomplish with so small a force?" Alvarez asked. "Suppose you are right, and you find not only the weapons, but an active base crawling with PAIN terrorists? What then, Ms. VanDeen?"
"Then we'll _know,_ General Alvarez. This isn't a military expedition, it's simply a scouting party. And if we don't come back by tomorrow evening, you'll know that we ran into trouble. If we do come back, we'll be able to tell you if the weapons are still there."
"It still strikes me as a waste of resources," said Alvarez. "But they aren't _my_ resources." He looked at Mingus.
"Gloria," Mingus said, "do I understand you correctly? You are planning to make this journey _personally_?"
"It's my job," she replied. "I mean, isn't this the kind of thing that you created OSI to do in the first place?"
"Broadly speaking, I suppose so," said Mingus. "Nevertheless, I never intended for you to go charging off like Custer looking for Indians. If you want to send your Internal Security people on this mission, I have no objection. But I think you should stay here."
"Stay here doing _what_?" Gloria demanded. "Looking pretty at parties?"
"Let her go," Cornell DuBray said from the other end of the table. Gloria looked at him in shocked surprise. He gave her a thin smile and said, "One thing I have learned recently is that it is never wise to underestimate Ms. VanDeen."
Mingus's eyebrows rose for a second, then he sighed and said, "Very well, then. Good luck, Gloria."
JILL AND ELI OPATNU TRUDGED DOWN THE LONG flight of marble steps leading from the Imperial Court. "You weren't much help," Jill said to him.
Opatnu spread his arms in wounded innocence. "What would you have had me do?"
"You might have backed me up when I said that it was critical to get the Wendover records immediately."
"What's so critical about it? Jill, this investigation is likely to take months, if not years. The judge isn't going to let us stampede him."
"But the longer it takes us to get those records, the more chance they'll have to alter them."
"They could do that in fifteen minutes," Opatnu pointed out.
"Well, we'd already have them if you hadn't lured me to lunch, then back to your hotel, that first day. If I had been alone, I'd have gone straight to court and I'd have gotten them before Wendover could have filed for a restraining order."
Opatnu stopped at the base of the steps and looked at her. "Are you telling me that you regret the hours we spent together in my hotel room?"
Jill gave him a crooked smile. "Well, no, I'm not saying _that._ "
"So if you're not complaining about that, what _are_ you complaining about? Last night?"
Jill looked down at her toes for a moment. "I just wasn't expecting to see you there. It caught me off guard."
Opatnu put his arm around her, drew her close, and gave her a kiss on the cheek. "I would never do anything to hurt you," he said. "Gloria and I weren't expecting to see you there, either."
She pushed away from him. "What you and Gloria do is none of my business. I'm sorry, Eli, I know I'm being foolish."
"What do you say we spend the rest of the afternoon being foolish together?"
"Well," Jill said after a pause, "I can't do anything more about Wendover. And I'm finished helping Petra. I suppose I really ought to spend the afternoon back at the Convention Center, schmoozing."
Opatnu seized her hand, brought it up to his mouth, and kissed it. "Schmooze _me,_ " he whispered.
And so she did.
PETRA ALLOWED HERSELF TO BE TAKEN TO lunch by Elizabeth Irons and some of her Internal Security people. It was a short lunch, since everyone but Petra had something to do, but Irons had been effusive in her praise. It was nice to be appreciated, and not just for her work; one of the young men on Irons's staff had spent a lot of time looking down her unbuttoned shirt and asked if she was free for dinner. Petra was flattered, but was just too worn-out by her labors to accept. After lunch, she returned to the Imperial Cantabragian and collapsed on her bed.
An hour later, she was awakened by the arrival of two detectives from the Central Police. Blearily, she led them into the main room of the suite and offered them coffee. They accepted, so she called room service.
She tried to stay focused as the detectives explained that they were here for a routine follow-up in their investigation of the murder of Jamie Quincannon. They mainly seemed interested in whether she had anything new she might share with them. They asked the same questions several times in slightly different ways, and Petra managed to answer them without saying much of anything. The coffee arrived and she perked up a bit.
"Dexta's investigation of the information we received from Mr. Quincannon is essentially complete," she told the detectives. "I really don't think we have anything relevant to add to what we've already told you. We already mentioned the possible zamitat connection. Didn't that lead anywhere?"
"No, ma'am," said one of the detectives, a man named Connors. "We haven't turned up anything that would point in that direction. Oh, we know Quincannon was a partner of Whitney Bartholemew, and if the old man were still alive, maybe the zamitat would make sense. But his kid isn't connected, as far as anybody knows. And it didn't look like a zamitat hit. They'd have shot him, neat and clean."
"So you don't have any leads in the case?"
"Didn't say that," said Connors. "Quincannon was mixed up in some shady real estate deals, and we're looking into that angle."
Petra nodded. "He tried to rent me a room," she said.
"He's got some property here and there around the city. We've checked up on some of his partners in the real estate deals, but haven't really developed anything. It's possible that it was just some low-level street crime kind of thing. That's a tough neighborhood, you know, and we didn't find any cash on Quincannon's body."
"He was a nice old man," Petra said. "I liked him."
The cops finally got up to leave, but Connors stopped. "Ms. Nash? I know this isn't any of my business, but half the people in the department are out looking for terrorists. I just wondered if you could tell me anything about how _your_ investigation is going. My wife is real worried and wants to send the kids to stay with her sister in Brattle."
"We think we're making some progress," Petra said, glad to be able to say that honestly.
"Like we're making progress on the Quincannon case? Or _real_ progress?"
"Real progress, we hope." She offered him an encouraging smile.
"Nice to hear that," said Connors. "I'll tell my wife to keep the kids here."
The detectives left and Petra sat down to pour another cup of coffee. It occurred to her that the two cops were about the only _real_ residents of Central that she'd met. And maybe Jamie Quincannon. Everyone else was some rich snob from up on the cliffside, or Dexta people. It was odd how you could spend weeks in a city and never really connect with anyone.
Her wristcom beeped. "Petra Nash," said Whit Bartholemew, "are you hungry? Or do you expect to be hungry at some point in the day? _I'm_ hungry. I hunger for the sight of your sweet face and flashing green eyes. I hunger for your wit and charm and intelligence. For your soft white flesh."
"Are you asking me to dinner, or have you taken up cannibalism?"
"The former, I assure you. I know your time here is limited, and the fireworks the other night prevented me from completing my wily seduction. In all fairness, you owe me another chance."
"Whit, I'm awfully tired. I was up half the night working."
"We'll make it an early evening, then. I'll pick you up at seven."
Petra shrugged. She was a single woman again, after all. Maybe an actual date was in order. And Whit Bartholemew was a man who intrigued her, although probably for all the wrong reasons. She decided abruptly that for one night, at least, the wrong reasons could be precisely the right reasons.
"I'll see you at seven," she said.
GLORIA, ARKADY VOLKONSKI, FIVE OF HIS Bugs, and a pilot accelerated away from New Cambridge. Within an hour of their departure, they would reach 92 percent of the speed of light, turn on the Cruiser's Ferguson Distortion Generators, and enter the strange realm of Yao Space. Interstellar flight was not very exciting, and the Bugs all tried to get some sack time.
The Cruiser could carry as many as twelve passengers in something approximating comfort. Two rows of bunks flanked a central passageway that connected the cockpit and a tiny galley with a small bathroom and the even smaller engine room. There was really not much need for an engine room at all, since in-flight repairs would have been virtually impossible if anything went wrong, but pilots found it comforting to think that they could control their own fate with a screwdriver and some elbow grease.
Gloria sat with Volkonski in the galley, sipping coffee and munching sandwiches. Volkonski had expressed his skepticism about this entire venture and, particularly, Gloria's participation in it. "If anything happens on that scumworld," he said, "you'll just be in the way."
"No, I won't!" Gloria protested. "You're forgetting, I commanded an entire army back on Mynjhino. I know how to handle myself."
"You commanded a bunch of half-assed volunteers, and nobody ever fired a shot," Volkonski said. "You've never been in actual combat. You haven't had any training in small-unit tactics. And you've probably never even fired a plasma rifle."
Gloria couldn't deny any of that. "So train me," she said. "We've got about thirteen hours. Tell me what I need to know."
Volkonski sighed. "It doesn't work that way, Gloria. We're not exactly set up to put you through boot camp here. It's not something you can learn from lectures or books. Anyway, if I tried to give you some instruction, you'd immediately decide that you were an expert and insist on running the show."
"No, I wouldn't."
"Of course you would. You couldn't help yourself."
Gloria scowled and propped her chin up on her fist. "You know me too damn well," she said petulantly.
"Keeping you alive requires a close study of my subject," Volkonski said. "I wasn't on Mynjhino, but I know what happened there and I know you tried to handle everything personally. And I saw you in action on Sylvania. Sooner or later, Gloria, you are going to have to learn that an executive simply issues orders and lets other people carry them out. I mean, why are you even _here_?"
"It's my responsibility to—"
"The hell it is. It's _my_ responsibility to carry out this mission. It's your responsibility to define the mission and see to it that I have all the resources necessary to complete it successfully."
"I think it's part of my responsibility to be with my people when I send them into a potentially dangerous situation."
"That's a very romantic notion," Volkonski said, "but it's also total nonsense. I mean, just look at yourself! Who the hell goes into a combat situation dressed like that?"
Gloria glanced down at herself. She was wearing a molecules-thick white bodysuit, set at 70 percent transparency, with the pressure seam in front opened from her neck to her navel. "I wore this on Mynjhino," Gloria said. "No one seemed to mind."
"I'm sure they didn't," Volkonski said, "because nobody on Mynjhino knew what the hell they were doing. But if we get into a fight on this scumworld, you'll just be a distraction to my men."
"I'll also be a distraction to the enemy."
"Or a target."
Gloria pursed her lips, then decided she might as well give in. She hit a contact switch and reduced the transparency of her bodysuit to 30 percent, then closed the pressure seam up to the base of her sternum. "There. Is that better?"
"Not really," Volkonski said, shaking his head. "But I suppose it's the best I can expect. Why must you always be the center of attention?"
Gloria reached across the table and put her hand on Volkonski's. "Arkady," she said, "I _like_ being the center of attention. And I don't really have any choice, do I?"
Volkonski looked at his beautiful boss for a few seconds and broke into a grin. "I suppose not," he said.
"I know how I look," Gloria said. "I realized early on that I could either fight it, and try to be drab and plain even though I wasn't, or just accept it, enjoy it, and try to get maximum mileage out of it. Dressing the way I do gives me a kind of power that isn't available to all you big, strong, tough guys. And I use that power to get the job done."
"You certainly do that," Volkonski agreed.
"And as for why I'm here, well, that sort of goes with the territory. Gloria VanDeen can't be just another bureaucrat. She's got to be out there where people can see her, sexy and glamorous to the end, even if it kills her. Spirit, do you know how I spent the last two weeks? I put myself on display and let myself be ogled and groped by half the bureaucrats in the Quadrant. And do you know what the result of all that was? OSI is going to make it, Arkady! We really are going to become the Fifth Quadrant! You saw it for yourself at the meeting this morning. OSI was right there at the same table with all the grown-ups. The public is crazy about me, I've gotten the support of the Dexta rank and file, and even people like DuBray have to accept the fact that OSI is here to stay."
"Unless," Volkonski pointed out, "Gloria VanDeen gets her beautiful head blown off on that scumworld."
"Well, yes," Gloria conceded. "There's always that, isn't there?"
WHIT BARTHOLEMEW FOUND HIS MOTHER SITTING on a sofa in the library of the Bartholemew family mansion on the cliff north of Central. Sitting with her, to his considerable surprise, was his grandfather, Norman Mingus.
Whit stopped short a few feet away from them and stared at Mingus. "What are _you_ doing here?" he demanded.
Mingus got to his feet, walked toward his grandson, and held out his hand. "Good to see you again, Whit. You're looking well." When Whit made no move to offer his own hand, Mingus withdrew his. "Still the same as I remember you, I see."
"Believe it." Whit turned to look at his mother. "What is this bastard doing here?"
Saffron Mingus Bartholemew waved her hand dismissively. "Oh, relax, Whit," she said. "It won't kill either one of us to spend a few minutes with him. Of course, a few minutes is all he has."
"Ah, the doting grandparent! Managed to find ten minutes for us after twenty years, did you?"
"Fifteen, actually," said Mingus. "But perhaps I was being overly optimistic."
"You can drag your ancient ass out of here right now, for all I care."
"Oh, now both of you, just sit down and stop acting like babies. Father, right here where you were. Whit, plant yourself in that chair and try not to suck your thumb and pout." Saffron stared at both of them, and the force of her gaze finally drove them to do as she had ordered.
"Father was just explaining why he couldn't come and visit us while Bart was still alive," Saffron said.
"And I was hoping that you might explain why you and Whit never came to visit me in Manhattan," said Mingus. "It was never my desire or intention to be alienated from you."
"Your desire and your intention," echoed Saffron. "You never really managed to sort those out, did you, Father?"
"You know why I did what I did," Mingus answered softly. "And it wasn't such a bad life, was it? Bart took care of you and tried to do right by you. He was not a bad man, was he?"
Saffron took her father's hand and squeezed it gently for a moment. "No," she said, "he wasn't."
"He was an even bigger bastard than his father-in-law," Whit interjected. "You ran your big noble Empire, while he ran his greasy little empire. And Mother and I were caught in between, and nobody ever gave a damn about that."
"That's not true, son," Mingus said. "I cared very much about you and your mother, and I'm sure that Bart did too, in his way. But life is never as simple as we would wish."
"Spare me the grandfatherly philosophy about life, old man. You want to know what life is? I'll tell you what it is. It's an eternal struggle between the fat cats like you and dear old Dad, and all the rest of the poor, stupid, swarming mass of humanity. Only they don't have a chance, do they? Because the fix is in. The game is rigged and always has been. People like you get what they want, and everyone else has to settle for wanting what little they get."
"You sound just like you did the last time I saw you," Mingus said. "You were just out of college then, with your long hair and your beard, spouting leftist drivel as if you were the first one ever to think of it. The boy's hardly grown up at all, has he?"
"Oh, that's not true," Saffron protested. "He's done a fine job of running the family businesses, and without any help from—from _those people._ You realize, don't you, that by the time he died, Bart had made his enterprises almost entirely legitimate?"
"Which is more than anyone can say for you," Whit said accusingly. "The Empire is the most corrupt enterprise in the history of humanity."
"The Empire"—Mingus sighed—"is what it is." He got to his feet, and Saffron and Whit did, as well. "I'm sorry, but I didn't come here to debate political theory. I have to be going, but I expect to see you—both of you—at the OSI reception Saturday night. You'll be there, won't you?"
"I don't know," Saffron said. "Of course, the Ellisons will expect us, but I just don't know."
"Come," said Mingus. "I'll introduce you to the Emperor. Whit too, if he'll promise to behave himself." Mingus kissed his daughter lightly on her cheek, paused to gaze for a moment at his grandson, then strode briskly from the room.
"He'll introduce us to the Emperor," Whit said with bitter sarcasm. "I have a much better idea. I've been planning a little working vacation, and I want you to join me. We'll leave Saturday afternoon and go to the mountain lodge on Belairus. You always liked it there, and the mountain air will be just the thing to clear the stench of all these bureaucrats."
"Belairus is lovely," his mother agreed. "But I honestly don't see how we can avoid that reception. The Ellisons would be disappointed and despite your disapproval, I suppose I _would_ like to meet the Emperor."
"I'll make you a deal, then," Whit said. "We'll go to that reception Saturday night, and I'll behave myself. Then we'll leave Sunday morning."
"That's still rather sudden. Couldn't we make it next week?"
"No," Whit said flatly. "Sunday morning. I told you, it's a working vacation, and I already have meetings scheduled. We can't leave any later than Sunday morning. If you don't agree, I'll make an ugly scene with the Emperor."
Saffron looked at her son. "You'd do that, wouldn't you?"
"Damn right I would. Sunday morning?"
"Very well, then. Sunday morning." Saffron pursed her lips and shook her head as she looked into her son's eyes. "I never could say 'No' to you and make it stick."
"Really? I always thought it was more the other way around. I remember one time when—"
Saffron pressed her fingers against Whit's lips. "Hush," she whispered. "We'll speak no more about it. Now, as for this evening, I had hoped we could have a—"
"Sorry, Mom," Whit said. "Got a date."
Saffron raised an eyebrow. "A date? Anyone I know?"
"You met her at the Ellisons' party. Petra Nash."
"That cute little girl who was with young Palmer Ellison? Yes, I gather they've broken up. Only to be expected, of course. Steffany is a much more suitable match for the boy. The Ellisons would never have consented to his staying with someone so...common."
"She's not common. In fact, she's delightfully uncommon."
"Oh? What's this? Honest attraction, dare I hope?"
"Decide for yourself. I'm going to ask her to join us on our trip."
"Will wonders never cease?" Saffron asked. "You aren't finally falling in love, are you?"
Whit had no answer for that. His mother thought he even looked a little embarrassed.
PETRA FELT EVERY EYE ON HER AS SHE WALKED into the fancy restaurant on Whit's arm, wearing only a filmy green band skirt, adjusted shockingly low on her hips, and a matching halter top that didn't quite conceal her nipples. Pug had gotten the outfit for her on their shopping spree, and while she doubted that it was something she would wear back home in Manhattan, it somehow seemed just right for this staid, heavy planet. One more way of thumbing her nose at them all.
They were seated at the same table they'd had two nights earlier, with its sweeping, panoramic view of the city. Wine was served, Whit approved it, and they raised their glasses in a toast. "Here's to two people in a trillion, finding each other in the midst of all this rot and decay," Whit said.
"You're cynical even when you're being sentimental," Petra noted.
"It's the Hegelian dialectic," Whit explained, "the eternal clash of opposing forces, out of which the future is forged."
"That's a little better," Petra said, and sipped her wine.
"With the right motivation, I could do _much_ better. You bring out the poet in me, Petra Nash. The other day in my office, we were simply two rutting animals. Under the proper circumstances, in the right setting, you might find that I can be quite human."
"I think you're more human than you realize, Whit. You're a very attractive man. Maybe I was just getting back at Pug the other day in your office, but that doesn't mean that I don't feel something for you."
" 'Something,' " Whit said. "Could you be more specific?"
Petra smiled. "Not just now," she said.
"Well, I feel 'something' for you, as well. I can't give a name to it, and I won't insult us both by pretending that it's love, but whatever it is, it's real and deeply felt."
Petra's smile turned into a grin. "Took the words right out of my mouth," she said.
Whit returned her grin. "We understand each other, then."
"Not as well as I'd like to. I wish I knew where all your anger comes from. It can be a little frightening."
He drank some more wine. "My anger," he said, "is my oldest companion. A friend of my youth."
"Have you ever tried living without it?"
"No," he said, "I haven't, and I have no wish to. My anger is my magnetic north. It provides orientation and direction in my life. It tells me what I must do, and why. Don't you ever feel it?"
"Not the way you do," Petra replied. "Sure, I get angry sometimes. But I don't let it control my life."
"What does control your life, then? What do you _want,_ Petra Nash?"
Petra sipped some wine. "That's not an easy question to answer," she said as Whit poured more wine for both of them.
"It never is," Whit said. "That's why we so seldom ask it of ourselves. Confronting our deepest desires is a risky business. We might learn something about ourselves that we'd rather not know. Take you, for example. You've just traded in an upright, respectable young boy for a ne'er-do-well, middle-aged rascal. Have you considered just why you did that?"
"It wasn't exactly something I planned, you know."
"Plans have nothing to do with it. The fact is, you did it. Here we sit, Petra Nash, together and, to all appearances, enjoying it. That is the reality of the moment."
"If it's reality, does it have to have an explanation?"
For a moment, Whit had no response. They watched in silence as a brightly lit vessel in the harbor rose into the dark sky on a plume of blue fire.
"I'm taking my mother away to our lodge on Belairus," Whit said, looking Petra in the eyes. "We leave Sunday morning. Come with us."
"What?" Petra asked, taken aback.
"You heard me. It's only a two-day journey in my yacht."
"Whit, I have a job. I can't just take off at the drop of a hat."
"Why not? The Quadrant Meeting will be concluded by then. You don't need to stick around just to watch the Emperor blab to the masses in the soccer stadium on Sunday. Don't you have some time off coming?"
"Well, as a matter of fact, I do," Petra said. "But—"
"Then take it now—with me! We have a beautiful lodge in the mountains. Have you ever seen Belairus? The most beautiful little world you can imagine. And low-gravity, too—only .82 G. You'll feel as if you were inflated with helium."
"Now _that_ sounds good!" Petra had already kicked off her high heels beneath the table.
"Then you'll come?"
"Spirit, Whit, I don't know what to say! There's so much going on, and even with the Quadrant Meeting ending, Gloria may need me to help wrap up some loose ends. I just don't see how I could leave Sunday. But I could talk to Gloria, and maybe next week, when she goes back to Earth, I could get away for a week or so."
Whit shook his head. "No, it must be Sunday."
"Why can't you just delay it a few days?"
"I'm sorry, but I can't. It must be Sunday. Talk to Gloria about it tomorrow."
"Can't. Gloria's not here and won't be back until tomorrow night."
"Not here? Where else would she be?"
"I can't tell you that," Petra said. "But it's important. Whit? I did something good today. Maybe something very good. You asked me what I want? I think what I want is the chance to do something like that every now and then. The chance to make a little difference in the galaxy."
"A little difference in the galaxy?" Whit smothered a chuckle in some more wine. "Why not a _big_ difference?"
Feeling slightly offended by Whit's reaction, Petra replied, "And what difference have _you_ made? What difference will you _ever_ make?"
"You never know, Petra Nash," Whit said, "you never know."
Later, Whit took Petra back to his big mansion, up a lavish staircase, and into his bedroom and bed. Petra wriggled out of her minimal clothing, lay back, and shuddered happily as Whit made his way down the length of her body, from the ticklish hollow of her throat to her straining pink nipples, her smooth, slim belly, and the warm, tingling wetness of her groin. He was confident and thorough, and left her gasping in urgent fulfillment.
Pug had been an ardent and playful lover, and she'd had no complaints, but there had always been something about him that made her think of a boy who thought he was getting away with something—or maybe, not quite getting away with it. It was as if, she thought, he could never really get out from under those murals staring down on him from the ceiling above. But Whit was sure of himself, selfish and, yes, angry. She could feel his anger as he entered her, plunging and pounding himself against her like a wave beating against a shore it desperately desired. A meeting of opposites.
And when he was finished at last, and lay next to her, panting and drained, he said, "Sunday morning. You'll come away with me. You must, Petra Nash; you _must_!"
GAC 4367-III WAS AN UGLY WORLD. FROM POLAR orbit in the Cruiser, Gloria and her Bugs watched the pea-green landscape slide by beneath them and she felt no desire to descend into that miasma. Here and there, a crenellated range of stunted mountains provided literal relief from the featureless continental masses, and ocean currents had cleared narrow, open arcs of the biological infestation, but all else was algae.
The terraformers of the old Terrestrial Union had believed that with the proper encouragement, a scumworld such as this might be turned into a garden. A billion years' worth of decayed algae would provide organic nourishment for muscular, better-evolved terrestrial plants, which would simply take over the planet. Sow some wheat, then step back and watch it grow, free of pests and competition. It had actually worked that way on a few worlds, but more often, the unexpected complications of alien ecosystems frustrated the hopes of the terraformers. GAC 4367-III occupied the attention of Earthbuilders for half a century, until better and easier worlds became available, and the experiment was abandoned. The triumphant algae reclaimed their world from the alien invaders.
"There's the terraforming station," said Volkonski as he and Gloria examined a display screen in the cockpit of the Cruiser. He fiddled with some controls and magnified the image until the sloping rooftop of what seemed to be an immense warehouse stood out against the background slime. The structure seemed to be covered with the algae mats, and only the long shadows of early morning defined its shape and size.
"I don't see any sign of people," Gloria said.
"I don't know," said Volkonski. "We're getting a faint infrared signature. That building is leaking some heat."
"Could it be from an old reactor?" Gloria wondered.
"Possible. They used uranium reactors in those days. It might still be generating a little heat. We should check for radiation before we enter that building."
"That looks like a dock on the shore of the bay."
"Right." Volkonski turned to the pilot, a sandy-haired young man named Erskine. "What do you think?"
Erskine shrugged inconclusively. "Spirit knows how deep that water is. There's no channel marked out that I can see. We'd have to land well offshore and hope for the best."
"And if anyone is down there," Volkonski said, "they'd have plenty of time to see us coming. I'd prefer to wait and make a night landing, but it's early morning at this location. Planetary rotation period is twenty-eight hours, so we'd have to wait fourteen or fifteen hours before we tried it."
Gloria shook her head. "We can't do that. We'd be half a day late getting back to New Cambridge. They might get worried and send another ship. Anyway, our main priority is to find out if the weapons are there and, if they are, to see if any of the big plasma bombs are missing. If they've taken one of them to New Cambridge, they could use it as soon as the Emperor arrives, tomorrow night. We have to get down there now, Arkady."
"Agreed. You heard the lady, Erskine. Set us down."
"Yessir," said Erskine. "But before I do, there's one more thing you should be aware of. According to these readings, the atmospheric oxygen content is 29 percent."
"Spirit!" Volkonski exclaimed.
"What's the problem?" Gloria asked. "We can breathe that, can't we?"
"We'll probably get drunk on it," said Volkonski, "but yes, we can breathe it. That's not the problem."
"Then what is?"
"Begging your pardon, ma'am," said Erskine, "but with 29 percent oxygen in the air, that planet is a fucking tinderbox."
"Oh?"
"Yes, ma'am. See these brown, splotchy areas scattered around? I think those are burned-out areas, probably from lightning strikes. Any sort of flame would immediately set off a conflagration. Anything that could burn—like the algae mats— _would_ burn, until rains put it out."
"What does that mean for us?" Gloria asked.
"For one thing," said Volkonski, "it means this entire planet is most definitely a no-smoking area."
"Yessir," said Erskine. "And it also means that our exhaust could start a fire."
"But other ships have obviously landed without burning down the planet," Gloria pointed out.
"Yes, ma'am. I think we'll be okay if we come down in the water just offshore. You can see that the wave action has pretty well broken up the algae mats. I don't think we'd touch off any big conflagration. We'll use thermoelectric propulsion once we're in the water, so that should be okay. But we need to be aware that there's a potential problem."
"Thank you for mentioning it, Erskine," Gloria said. "Anything else we should be worried about?"
"Rocks just under the water, ma'am. Once we're down, we'll be able to spot them with our sonar. But the actual touchdown will be a crapshoot. Come down in the wrong place and we could rip the bottom out of our hull."
Gloria sighed in frustration. "If only they'd given us a LASS. It never occurred to me that there could be so many complications in just landing on the damned planet."
"That's the problem with ordering these romantic little excursions," Volkonski said. "They're never quite as easy as they sound when you're giving the orders."
" 'Ready, fire, aim,' huh? Sorry, Arkady, I guess I should have thought this through a little better."
"If you had," he said, "we'd be here anyway, doing exactly the same thing. You were right, Gloria. This is a job that has to be done."
"Then let's do it," she said. "Erskine, let's shoot some craps."
ERSKINE BROUGHT THE CRUISER IN FOR A feather-soft touchdown a mile offshore. As he carefully puttered the craft in toward the dock, Volkonski focused the image-intensifiers on the area and warily scanned the screen.
"Someone has cleaned off that dock and the area around it," he said. "People have definitely been here, and more recently than eight centuries ago."
"If anyone's here now, they would have seen or heard us coming in," said Gloria. "But I don't see anyone near the dock."
"They could be hiding behind those rocks, or farther inland. If I had my druthers, I'd send in a boat with a few men to cover our landing."
"Why don't you?"
"We don't have enough men to divide our force that way."
"So I should have sent a bigger team?"
Volkonski shook his head. " _I_ should have. Tactical considerations are my job, not yours. Speaking of which, I should go aft and get the squad ready. Keep a watch and sing out if you see anything."
Gloria stared intently at the image screen but saw no movement of any kind. There were no birds or insects, no creepers or crawlers. It was a one-celled world.
"How are we doing, Erskine?"
"So far, so good, ma'am. Water depth is okay, and the bottom is sandy with a gentle slope. Those Terrestrial Union guys probably surveyed the area and built their dock in a good spot. We should be fine."
They were. Erskine expertly guided the vessel toward shore and brought it in snug against the dock. Gloria patted him on the shoulder, then went aft to join Volkonski and his Bugs.
The five IntSec men and Volkonski wore their standard gray uniforms, pantlegs tucked into black boots, with bulky, glossy helmets packed with electronic gear. Each of them hefted a Mark VI plasma rifle, except for Volkonski, who carried a holstered plasma pistol.
"Minimum beam setting on all weapons," Volkonski instructed. "No one fires at anything, for any reason, without my specific order. And if you do have to use your weapons, try to hit the person you're aiming at and nothing else. And under no circumstances does anyone fire in the direction of that building. Is that clearly understood? We don't know what's in there, but if you put a hole in the containment of a plasma bomb, your mothers, wives, and/or sweethearts would be very upset. Any questions?"
"Where do you want me?" Gloria asked. "Right behind you?"
"I want you exactly where you are now. In the ship, buttoned up."
Gloria started to protest, but one look from Volkonski closed her mouth. "I'm sorry, Arkady," she said. "You're in charge. I'll do as you say. But I want you to keep an open comm link at all times."
"Will do. And Gloria? If things go wrong out there, you and Erskine get the hell out of here immediately, understand? Someone can come back for us later, but the vital thing is to get word back to New Cambridge."
Gloria nodded. Volkonski gave her a stern look, as if to reinforce the message, then, without another word, turned and hit the control that opened the sliding hatch panel. An airlock was not necessary, since a mass-repulsion field kept the interior and exterior atmospheres separate—at least in theory. But somehow, a pungent waft of the scumworld managed to penetrate into the Cruiser.
"Phew!" Gloria cried.
"Hydrogen sulfide from rotting algae," Volkonski said. "Okay, take a deep breath and let's go!"
The Bugs charged out onto the dock, each of them running to a specific spot, as if the whole operation had been choreographed in advance. They knelt and scanned the horizon as Volkonski went out at a dogtrot to the landward end of the dock. Then the first men out ran past Volkonski and stationed themselves in the jumble of rocks onshore. One of them immediately slipped on the slimy algae mats and took a header. He picked himself up, looking chagrined, and Volkonski called out, "Watch where you step!" He looked back at the Cruiser and motioned for Gloria to close the hatch. She watched the Bugs advance for another moment, then reluctantly hit the control to seal up the ship.
Gloria went forward, where Erskine had already established the comm link with Volkonski. The image on the screen bobbed up and down and from side to side as Volkonski's helmet moved, and she could hear his heavy breathing. The land sloped upward from the water, and the big building, less than half a mile away, could not be seen from Volkonski's current location. The Bugs fanned out, slipping and sliding on the algae as they moved. It was not a graceful-looking operation, but the team made steady progress inland.
"Iglesias! Reynolds!" Volkonski shouted, and the screen showed his arm extended, pointing to locations on either side of what seemed to be a pathway leading upward. The two men hustled to the spots at the top of the rise and threw themselves onto their bellies. Erskine hit a control button, and suddenly Gloria was looking at the scene from Reynolds's viewpoint. Ahead, the massive structure built by the terraformers of long ago loomed on the horizon. It seemed to have been constructed from sheets of corrugated metal, and there was a large central doorway, currently closed. The entire building was coated with the omnipresent gray-green slime.
"I got movement!" shouted Iglesias. Erskine switched to his camera, but Gloria saw nothing amiss.
"Here, too!" called Reynolds. Now, Erskine called up a split-screen image, and they could see the view from the positions of both Bugs. Gloria thought she could see a flicker of motion near some rocks in the mid distance.
"I make it three...check that, four unknowns, at three hundred meters," Reynolds reported. "They've got some kind of camo on, hard to see."
"Visors," Volkonski ordered. Immediately, the view on the screen switched to the ghostly, green glow of infrared imaging. Now Gloria could plainly make out at least three human forms crouching near some boulders scattered to the front. Volkonski advanced and took a position next to Reynolds. Gloria saw his hand at the edge of Reynolds's screen, his index finger pointing toward a little swale ten meters ahead. As Reynolds scrambled forward, there was a muffled bang from somewhere, and the image from the Bug's camera suddenly showed nothing but algae and mud.
"Shit! I'm hit!"
"Stay down!" Volkonski shouted. "Byerly, Mitsui, go get him! Gordon, Iglesias, covering fire! Over their heads! I don't want to start any fires, but let's show 'em we've got some teeth!"
The dazzling bursts of plasma overloaded the imagers for a moment. To the sound of grunts, heavy breathing, and a pained moan, Byerly and Mitsui dragged Reynolds back to the near side of the rise. Volkonski knelt over his man, and the screen showed Reynolds's anguished face and then the ripped fabric of his uniform and the dark stain spreading from his right shoulder.
"Flèchettes," Volkonski muttered. "Figures—they don't want to start any fires, either. Relax, Reynolds, it probably hurts like a son of a bitch, but it's not that bad. You're going to be fine." Volkonski looked around. "Gordon, Iglesias! What do you see?"
"Still just the four of 'em, sir! Not moving."
"Okay, keep watching, and if they try to move, fire more warning bursts. Gloria?"
"Right here, Arkady."
"I think we've got an answer to our main question."
"I think you're right. What do you recommend?"
"Under the circumstances, our plasma weapons are useless, but they can use their flèchettes. I think we should pull back to the Cruiser and get the hell out of here."
"Understood," Gloria replied. She still wanted to get into that building and see if any plasma bombs were missing, but that no longer seemed possible. Arkady was right: Their only mission now was to get back to New Cambridge and report what had happened.
Volkonski issued the necessary orders, and Byerly and Mitsui carried Reynolds back to the dock, while Gordon and Iglesias remained at the top of the rise to provide covering fire if necessary. Then he ordered Gordon and Iglesias to fall back. Gloria opened the hatch and stood to one side as Byerly and Mitsui came in with Reynolds and gently placed him on one of the bunks. The young man was alternately gritting his teeth and gasping for breath, but when he saw Gloria, he tried to smile. She returned the smile and squeezed his hand. It was covered with slime.
"Sorry 'bout that, ma'am," he said. Gloria blinked back tears and said nothing.
A moment later, Volkonski, Gordon, and Iglesias came tumbling into the Cruiser. Volkonski smacked the hatch control with his palm and shouted to the pilot, "Get us out of here, Erskine!"
The bass rumble of the engines vibrated through the ship, and the vessel gently lurched into motion. Then Erskine cried, "We're under fire!" Volkonski and Gloria dashed forward.
"Plasma burst into the water," Erskine said. "Damn, another one!"
Volkonski checked the external imagers. "And one behind us, too. Hold where we are, Erskine."
"Yessir!"
"Two of them," said Volkonski as he examined the screen, "up there at the top of the rise."
"What can we do?" Gloria asked.
"Not one damn thing," Volkonski said between his clenched teeth. "They want the ship. They could put a hole through the hull anytime they want and prevent us from taking off, but they haven't done that. They want us to stay here and they want the ship intact."
"Can't we fire back?"
"No external weaponry on this tub. We could open the hatch and start a fire out there with a plasma burst, but they'd still have time to burn us. What we seem to have here is an old-fashioned Mexican standoff."
"Can we talk to them?" Gloria wondered.
"My thought exactly," said Volkonski. "Erskine?"
"External mikes and speakers on, sir. Just use your throat mike."
Volkonski took a couple of breaths, then said in an authoritative voice, "This is Dexta Internal Security. Cease fire at once!"
"Fuck you, Bug!" came the reply, followed by another blue-green plasma discharge into the water just ahead of the Cruiser.
"Who are you?" Volkonski demanded.
"We are the People's Anti-Imperialist Nexus, and we've got you in our crosshairs, lickspittle!"
"I guess you boys don't get many newstexts out this way," Volkonski said in a more conversational tone. "PAIN's leadership group has been captured, and the whole operation is disbanding. The war's already over, and your side lost. Give it up now and you'll get off easy."
"You must think we're idiots!"
"As a matter of fact, I do. Who else but idiots would let themselves get stranded on this stinkhole fighting for a cause that's already lost? How long have you guys been stuck here, anyway?"
There was no immediate response. Then a different voice called out, "None of your business, Bug!"
Volkonski put his hand over his throat mike and said to Gloria, "I thought so. They've been here a while. The main thing they want is to get off this planet. I don't think they'll do anything to damage the ship."
The first voice from outside called, "Hey, Bug! We'll make you a deal. Come out with your hands up, and we won't kill you."
"We'll make _you_ a deal," Volkonski replied. "Put your weapons down and we won't set fire to your planet. Twenty-nine percent oxygen out there—you'd go up like torches."
"Then we'll all burn together, Bug," came the response.
"He means it," Volkonski said to Gloria.
"This could go on all day," Gloria said.
"Unless you have a better idea."
"Just one," she said. "I think this may be the time for a distraction." She hit the contact switch that quickly rendered her bodysuit 90 percent transparent, then opened the pressure seam in front as low as it could go.
"Just remember the mission, Arkady. If you get a chance to get away, take it. You can always come back for me later."
Volkonski nodded. "Good luck, Gloria."
Gloria leaned toward Volkonski's throat mike and announced, "This is Gloria VanDeen of Dexta. I'm coming out."
WHIT BARTHOLEMEW'S LIMO RETURNED PETRA to the Imperial Cantabragian that morning, and she walked through the crowded lobby, still nearly nude in her flimsy nighttime togs. She immediately encountered Althea Dante, who gave her a friendly kiss on the cheek and gushed, "Petra, darling! We're all so proud of you! A marvelous piece of detective work."
"They made a public announcement?" Petra asked.
"Of course not. But Dexta people know. And don't you look glamorous!" Althea gave her a wicked grin. "I trust you had a fulfilling evening?"
"It was interesting," Petra said.
"I'll just bet it was. You're getting to be quite the Tiger, aren't you? Maybe I could teach you a few things sometime."
"C'mon, Althea, you know I don't do that."
"Variety is the spice of life," Althea countered.
"I think I've got enough spice in my life for now," Petra said. "But tell me something. What do you know about Whitney Bartholemew, Junior?"
"Not much," Althea replied. "I get the impression that you must know a lot more than I do. Why do you ask?"
"Just wondering. He's a strange man, in some ways."
"So I gather. I never really knew him, but as it happens, we went to the same college. He was a few years ahead of me, of course. At the time, he was one of those hairy campus radicals—you know the type. Always hogging the microphone at demonstrations and droning on and on about the evil pigs and the Hagoolian dialectic."
"Hegelian," Petra corrected.
"Whatever. He was never exactly my type." Althea eyed her inquisitively. "Is he yours, Petra, dear?"
"I'm not sure," Petra answered. "Althea, what do you do when you're attracted to a man you know is bad for you?"
"I usually give in to my own low urges and let the chips fall where they may. But that's just me. I'm not sure it's _you,_ though. Petra, would you mind some gratuitous advice?"
"Please."
"You're a delightful and charming young woman, and very sexy when you want to be. But you have an unfortunate streak of innate goodness in you. That can be very inconvenient when it comes to sex. When you get right down to it, women like Gloria and I are—to be coarse about it—cunts. The difference between us is that Gloria wants people to think that she's a _nice_ cunt, and I don't give a shit _what_ they think. But you really _are_ nice, in some fundamental way. Don't lose that, Petra. Just because things didn't work out with Peter Pan, don't throw yourself at Captain Hook. You're better than that. And now," Althea said, giving Petra another peck on her cheek, "I must be off. _Ciao,_ darling!"
Petra made her way to the elevators, deep in thought, pausing briefly to accept congratulations from two Dexta people she didn't even know. Once in the suite, she stretched out on a sofa and contemplated her navel, which she had heard was a good way to achieve enlightenment. She thought she needed some of that.
Whit wanted her to fly away with him on Sunday. With his mother. Just what she needed, another disapproving matron to impress. She wanted to go, wanted to spend a week or so on a far-off world—with low gravity!—and just forget about everything that had happened on New Cambridge. Lose herself in the angry passion of Whit Bartholemew.
But why did it have to be Sunday? Why couldn't he just wait a few days? What was so important about Sunday?
Abruptly, unexpectedly, Petra achieved the enlightenment she had sought. It had a physical force to it, and it almost made her ill.
She grabbed her purse and pad and all but ran out of the suite to the elevators. She had to get back to her office in Gibraltar. Enlightenment was one thing, but she needed facts.
GLORIA STEPPED OUT ONTO THE DOCK AND IMMEDIATELY wrinkled her nose at the pungent scent of the scumworld. The oxygen-rich air seemed to burn her lungs, but after a couple of deep breaths, she felt better. The air was cool and the sky was a faded blue-green. She saw two men flanking the path at the top of the rise, aiming their rifles at her. One of them carried a flèchette gun, the other an old Mark IV plasma rifle. She walked slowly to the end of the dock and waited. One of the men waved her on, and she made her way up the path; it looked as if someone had sprayed it with an herbicide that kept the surrounding algae mats in check.
Reaching the top of the rise, she paused again. Two more men appeared, one with a flèchette rifle and the other holding a plasma pistol. They were all dressed in pea-green jumpsuits, mottled with random patterns of brown and tan.
"Who's in charge here?" Gloria asked.
"No one," said the man with the pistol. "We are not hierarchical. We are an affinity group and we make collective decisions." Yet it was clear to Gloria that, anarchist ideology aside, the man with the pistol was effectively in charge. He was about thirty, brown hair, brown eyes, and like the others, wore a scraggly growth of beard.
"You mean I have to negotiate with all four of you at once?"
"Who said anything about negotiations?" said the man with the pistol. All four men were staring at her with what seemed to Gloria to be hungry appreciation.
"We didn't invite you to come out here," said the man with the plasma rifle.
Gloria slowly turned to look at each of them. "Come on, guys," she said, "we've got a situation here, and we need to discuss it. You obviously want our ship, or you'd have holed the hull by now."
"We can do more than that, Ms. VanDeen," said one of the men with a flèchette rifle. "We've got plasma RPGs, and we can blow that ship straight to hell."
"And probably yourselves, too," Gloria pointed out. "But even if you lived, you'd still be stuck here. How long have you been here?"
"Long enough," said the man with the pistol. "But our relief could arrive at any time. Maybe even today."
Gloria shook her head. "I wouldn't count on it. Didn't you hear what we said? PAIN is out of business."
"Why should we believe you?"
"What choice do you have? Look, we've got something you want, and maybe you've got something we want. Let's walk over to your building so I can get a look at those weapons—if you have them."
"We've got 'em, Ms. VanDeen," said the man with the plasma rifle.
"Then let's see them. I'm not going to negotiate until I see what we're negotiating about."
The man with the plasma pistol frowned, then turned to one of the men with a flèchette rifle and said, "Take her down the path a little way and watch her." Then he walked past Gloria and huddled with the other two men. Her guard motioned for her to move, and she did, but she noticed that the other three men were engaged in what seemed to be a heated discussion behind her. Apparently it was difficult for anarchists to give orders to each other.
The man guarding her was tall and thin, with dirty blond hair and hazel eyes. From the look of his beard, she guessed that they had been here at least three months. And they probably hadn't seen a woman in all that time.
Gloria smiled at him. "I'm Gloria," she said. "What's your name?"
The man hesitated, then answered. "Doug," he said. "And that's Marty, and the other two are Alex and Rick." Doug was trying to look Gloria in the eye, but his gaze kept darting down to her mostly uncovered breasts and her torso, which was not at all concealed beneath the clinging, nearly transparent bodysuit.
"Is it just you four?" Gloria asked. "Must get lonely around here."
Doug started to answer, stopped himself, then said, grinning self-consciously, "We got a whole army just over that hill."
Marty, the man with the pistol, returned, having completed his parley with the rest of the affinity group. His comrades looked back over their shoulders at Gloria, then turned to concentrate their attention on the Cruiser.
"Okay," Marty said, "walk. And if you try anything funny, we'll shoot you where you stand. We mean it."
"I know you do," Gloria said. "One of your comrades parted my hair with a plasma rifle a few weeks back."
"Yeah," Doug said, "we heard they were going to try for you. Kind of glad they missed."
"Shut up and walk," Marty said, and so they did. Gloria kept ahead of the two men, putting a little extra hip waggle into her stride. Anarchists, Gloria assumed, had the same hormones as other men.
After a few minutes, they reached the building. The main door, about twenty feet on a side, was closed. Marty indicated that she should go through a smaller door to the right. That led into a small pressure chamber, and when the outer door was closed behind them, there was a brief hissing sound and the inner door popped open. Inside, the stench was less noticeable, and the air seemed more Earth-like.
"Just so you'll know, we can use a plasma weapon in here," Marty told her.
"I'm sure there won't be any need for that," Gloria replied. Marty led her down a short corridor, opened a door, and gestured for her to go ahead. She stepped out onto the concrete floor of the main room. Ahead of her sat a medium-sized freight skimmer with a closed cab and a flatbed, parked immediately in front of the large pressure door at the entrance. Beyond it, she saw a variety of unfamiliar machinery and equipment, apparently left behind by the terraforming crew. To the right, and extending two hundred feet to the rear of the building, were stacks of shipping containers and crates of various sizes. Overhead panel lighting cast a diffuse, yellowish glow.
Gloria walked ahead of the men, toward the containers. She glanced around, like a potential house-buyer idly checking out the premises. When she reached the first of the containers, she paused and read the stenciled notations on their sides. Deciphered, the string of letters and numbers seemed to announce that the containers each held two orbital plasma mines. The date read 06-23-63.
"This what you came to see?" Marty asked, a hint of a sneer in his voice.
"Some of it," Gloria said, trying to sound unimpressed. "Where are the big plasma bombs?"
"Back there," Marty said, pointing with his pistol.
"How many?"
"Enough."
"Come on, Marty," Gloria said. "Don't play cute. We know there were eighteen of them in the original shipment. You sent one of them to New Cambridge, and we already have that one, so you should have seventeen left here. Unless you sent some of them elsewhere. We need to know. Seventeen?"
Marty pursed his lips for a moment, then nodded. "Seventeen," he said. "Back there on the left."
"Let's see them. I want to count."
They walked on. Marty said, "How do we know you really got the one on New Cambridge?"
"I told you, we broke up your whole New Cambridge operation. Think about it. How else could we have found you here?"
"She's telling the truth, Marty," Doug said.
"Maybe. So why'd they send you here, Ms. VanDeen?"
Gloria turned and smiled at the two men. "Oh, you know. Glamour. Publicity. 'Beautiful Dexta Agent Finds Missing Arms Cache.' That sort of thing."
"Figures," Marty snorted.
"You boys are just a loose end," Gloria told them. "Nobody from PAIN is coming to pick up you and the weapons. We're your only ticket out, guys. Or do you want to spend the rest of your lives here?"
"Shit, I knew it!" Doug said in disgust. "They said it would just be two months, and it's already been three."
"Shut up," Marty told him.
"Shut up, yourself! I'm sick of listening to your crap, Marty. First among equals, my ass!"
"Are these the plasma bombs?" Gloria asked, pausing in front of some big black containers.
"That's them. Go ahead and count, if you want. There are seventeen of them, like I said." Marty waved his pistol in her direction. "But don't get any stupid ideas."
Gloria made a show of counting and inspecting the containers, crouching and bending as needed to give Doug and Marty a good view of her assets. Her bluff was working, so far, and sex was providing the necessary distraction. It was simply a question of waiting for the right moment.
"Okay," she said at last, "seventeen. Maybe we can do some business, but I need to see the other stuff, too. Where are the plasma rifles and grenades?"
"Over here, on the right," Marty said.
Gloria walked over to the crates. "I don't want to have to count all 24,000 rifles," she said. "Help me out here, guys. How many of them have been distributed so far?"
"Couple hundred," Doug said, "give or take. About the same with the grenades."
Gloria looked at Marty. "That right, Marty?"
"Yeah," he said with some apparent reluctance. "That's about right."
"Good," Gloria said. "Now we can negotiate."
"What did you have in mind?" Marty asked her.
"We'll trade you. Our Cruiser for the weapons. You can leave and go anywhere you want. We'll stay here, and when we don't show up on time at New Cambridge, they'll send another ship for us." Gloria gave him a satisfied smile.
"We don't have a pilot," Marty said.
"You can have ours, as long as you promise not to hurt him."
She could tell from Doug's face that he was eager to accept the deal. But Marty looked as if he needed a little more persuasion.
"And I'll throw in a bonus, just for you two. The others don't need to know a thing about it."
"What bonus?" Marty demanded.
"How long since you guys have had a woman?" Gloria stood in front of them, grinning.
Doug's eyes bulged out, but Marty remained skeptical. "You expect us to believe you're going to put out for us?"
Gloria allowed herself to look wounded by this show of doubt. "Hey, Marty," she said, "I'm Gloria VanDeen! You've heard all about me, right? Sexually voracious. Wantonly uninhibited. The best fuck in the galaxy. And it's all true. But don't take my word for it—find out for yourselves! You must have some beds in this dump."
"Over there in the crew quarters," Doug said, trying to be helpful.
"Shut up. It's some kind of trick."
"Oh, fuck you, Marty! Three months in this fucking place, and for what? You heard her—it's over! This is our only chance to get out of here!"
"Maybe. But I don't like this sex business. Use your head, Doug. She's just trying to split us up."
"One of you can fuck me, and the other one can watch, with a gun in his hand," Gloria said. "I don't mind. Unless you think you wouldn't be able to...you know...do it. I mean, that happens with some guys. I think I must intimidate them or something. And, of course, if it's just been you four guys here together for three months, well, maybe you've worked out some other arrangement for yourselves. And that's okay, really. There's nothing wrong with same-sex sex. I mean, I've done it with plenty of other women, and if you guys are like that—"
"Hey!" Doug interjected. "It's not like that at all. You just let me show you!"
"You moron! Can't you see what she's trying to do?"
"What are you worried about?" Doug turned and got in Marty's face. "We've still got the guns! What can she do?"
At that precise instant, Gloria showed them what she could do. A flying Qatsima kick caught Marty square in the face and sent him reeling backwards, his plasma pistol clattering along the concrete. Doug reacted, but not quickly enough, and as he fumbled with the awkward flèchette rifle, Gloria bounded up in another kick that put a heel in Doug's groin. He doubled over, and Gloria finished him off with a knee to the chin.
Gloria picked up the plasma pistol and raised her left wrist near her mouth. "You hear all that, Arkady?"
"Every bit of it," Volkonski said over the wristcom. "Is everything secure there?"
"No sweat," Gloria assured him. "I've got the pistol, and Doug and Marty are in dreamland."
"Nice work, Gloria."
"Thank you, thank you. There's a freight skimmer, and I think I can use that. Just give me a few minutes to tie up the affinity group here."
"Roger. Nothing new happening here."
Gloria bent over and checked the anarchists' pulses, relieved to find that she hadn't killed either one. She looked around, and after a brief search found some bungee cords wrapped around the cartons of grenades. At summer camp, when she was eight, she had learned how to tie a variety of knots, and she used that knowledge now to secure Doug's and Marty's arms behind their backs and tie their feet together. Then, one at a time, she dragged them over the concrete floor to the front of the building next to the freight skimmer. Maneuvering them into the cab of the lorry took some doing, and she had to pause and catch her breath after the job was done.
She inspected the cab of the skimmer and found that she could operate it. Then she checked out the mechanism controlling the big pressure door and found that it involved no insoluble mysteries.
"Arkady? I'm all set here, but we need to coordinate."
"What are you planning?" Volkonski asked.
"I thought I could just barrel on out of here in the skimmer and down to the dock. If you can give me some covering fire..."
"It won't work, Gloria," Volkonski said. "Even if you got past the two guys on the rise, they'd still have plenty of time to put a hole through our hull. What's your range from the building to the rise?"
"Maybe five hundred meters," Gloria said. "Why?"
"Too long for accuracy with a flèchette. You'll have to use a plasma rifle."
"To do what?"
"To pick off the last two from behind."
"But wouldn't that start a fire?"
"Not if you're careful."
"Hey, Arkady, I've never even fired a plasma rifle. Anyway, I really don't want to have to kill anyone."
"Gloria," Volkonski said patiently, "this is no time to get softhearted."
"Give me a couple of minutes, Arkady. Let me think."
Gloria wandered around the vast shed, almost at random, pondering the possibilities. She came to an opened crate of plasma grenades, reached into it, and lifted one out to inspect it. It felt heavy in her hand, not the kind of thing you could throw very far. She knew there were slinglike launchers that were used to fling them a fair distance, but didn't see one. On the underside of the grenade, there was what appeared to be a timing mechanism. She studied it for a few moments, then went back to the skimmer.
She contacted Volkonski and told him what she had in mind. He didn't sound very enthusiastic about it, but didn't have a better idea. Reluctantly, he gave her the go-ahead.
Gloria got into the cab of the freight skimmer and fired up the engine. The skimmer rose a couple of feet off the concrete and hovered patiently. After checking to see that Doug and Marty were still unconscious, she got out of the cab and walked to the controls of the pressure door. She hit a couple of buttons and the inner door rose. Two more buttons, a harsh warning _blat!_ from a Klaxon, and, with a loud sigh, the outer pressure door lifted. Cool, putrid air rushed into the warehouse. Gloria dashed back to the cab of the skimmer and started it moving.
She paused just outside the building. After setting the timer on the grenade, she opened the door and tossed it out into the vast mat of algae. Then she shut the door and floated forward along the path. Ahead, she could see Alex and Rick, in apparent confusion, watching as the freight skimmer closed the distance between them. A hundred meters short of them, she paused and spoke into her wristcom. Volkonski had linked it to the external speakers on the Cruiser.
"Alex! Rick! Listen to me! This is Gloria, and I've got Marty and Doug here with me in the cab. They're still alive, and you guys can stay alive, too, but only if you do exactly what I tell you."
Gloria waited. The two men had pointed their weapons at the skimmer, but they looked back and forth at each other for a few moments before one of them shouted, "You'll never make it to the dock, lady!"
"Neither will you, if you don't shut up and listen," Gloria responded. "In about thirty seconds, a plasma grenade I left back there is going to explode. The door is open, and the fire will spread inside. The munitions in there will start cooking off, and pretty soon something's going to put a hole through the containment of one of those plasma bombs. You know what that would mean."
She paused for a few seconds to let them think about it. "All seventeen of them will go off, but one will be more than enough. In the meantime, that fire will be spreading in this direction. There's only one way off this planet, so if you want to live, drop your weapons and get down to the dock as fast as you can."
Before either man could reply, the grenade detonated. Gloria saw a green flash in her rearview mirror, then twisted around in her seat to get a direct view. She was amazed by what she saw.
In the oxygen-rich atmosphere, the plasma released by the grenade had touched off an instant inferno. Flames fifty feet high engulfed the warehouse. The mat of algae flared into brilliance and, as Gloria watched, the blaze moved rapidly in her direction. Too rapidly.
Gloria wasted no more time watching. She gunned the skimmer and dashed forward. Ahead, Rick and Alex had flung their weapons aside and were running toward the dock. A quick glance in the mirror told her that they would never make it, so as the skimmer reached the top of the rise, she slowed just enough for the two men to leap onto the flatbed. Gloria charged down the slope, turned sharply, and glided onto the dock. Before she could get out of the door on her side, two of the Bugs had opened the other door and were extracting Marty and Doug from the vehicle. As she darted around the front of the skimmer and headed for the hatch of the Cruiser, she took a last look over her shoulder and saw the blazing algae at the top of the rise. She dived into the Cruiser, the hatch closed, and Erskine maneuvered them away from the dock.
Gloria looked up and saw Volkonski standing above her, a half smile of admiration and amazement on his features.
"Whew!" she said.
"We're not out of the woods yet," he told her. He extended a hand, pulled her to her feet, then turned and went forward into the cockpit. Gloria followed.
"Erskine?" Volkonski asked.
"Workin' on it, sir!" Erskine replied breathlessly.
"Forget about rocks," Volkonski commanded. "Just get us _up_!"
"Yessir!" Erskine worked the controls so rapidly that even the sensitive mass-repulsion units that softened acceleration and provided ersatz gravity inside the Cruiser couldn't react fast enough. Gloria felt a lurch and almost lost her feet as the Cruiser clawed its way into the sky.
On the image screen, she saw the dock far below, quickly receding into invisibility. Beyond it, a flaming landscape and the warehouse, smoking and blazing. Then, even the warehouse was gone, and there was nothing left to see but the curving shoreline of the big bay—
And a flash that overloaded the imaging system for a moment. When vision was restored, she saw a billow of green fire rising from the receding landscape, and a concentric shock wave racing outward in every direction. She felt a slight bump as it caught up with the Cruiser.
Erskine looked around with a wan, weary grin on his face. "Made it!" he said.
"With six or seven seconds to spare," Volkonski said. "That was cutting it a little fine, Gloria."
Gloria put her fists on her hips and glared at Volkonski. "You're just never satisfied, are you?"
PETRA DOUBLE-CHECKED THE FINAL RESULT. Not because she wasn't sure—she had been sure ever since her moment of enlightenment back in the hotel—but because she needed to steel herself to do what had to be done next. She stared at the console for a long moment, then slid her chair back and got to her feet.
She walked out of her small office and down a corridor to the office of Elizabeth Irons, chief of Internal Security for Quadrant 4. Petra gave a nod to the assistant at the outer desk, but didn't pause, and barged straight into Iron's office.
Irons's mouth fell open slightly when she saw Petra in her scandalously revealing night-on-the town garb. "Ms. Nash," she said, "I don't know what they wear to work back in Manhattan, but here on New Cambridge—"
Petra interrupted her. "Ms. Irons," she said, "you have to arrest Whitney Bartholemew, Junior—immediately!"
PETRA STOOD ON THE SIDEWALK WITH Elizabeth Irons, a block from the Bartholemew Building in downtown Central, as Dexta, Imperial, and local security forces swarmed around the area. The immediate neighborhood had been evacuated, not that it would make any difference if a quadrijoule plasma bomb detonated. Somewhere high in the building, Whit Bartholemew was surrounded, with no way out, and Spirit knew what he might do.
She wondered how long she had known the truth and refused to recognize it. That moment of enlightenment in the hotel had not come out of nowhere. No one else had seen the truth, either, but no one else had been as close to the investigation or as close to Whit. She could hardly have been closer; just hours ago, he had been inside her, a welcome presence in her body and her life. Even now, she regretted that there would be no trip to Belairus, no more nights of angry lovemaking.
People stared at her, as if they knew what she had been doing; she felt their eyes on her all-but-naked body. She wished that she had taken the time to change her clothes in the suite, but her internal mood somehow matched her external appearance. She was emotionally naked, too, with no fig leaves of rationalization left to hide the truth: She had been passionately involved with a man who was a mass murderer. Something to put on her résumé.
"He wants to see you."
Petra noticed that Irons was staring at her. "What?" she asked.
"I just got word on the comm that Bartholemew wants to see you," Irons said. "He's holed up in his office, and apparently has a detonator switch that he says will set off the plasma bomb. You need to go up there."
"Yes, ma'am," Petra said. "I guess I do."
"A hundred million people could die if he sets off that bomb. We can't evacuate the city—it would be foolish to try. I'd give you a weapon of some kind, except that you have nowhere to hide it, and he'd probably insist on a strip search anyway. But if you get the chance to disarm or disable him..."
"I know a little Qatsima," Petra said. Very little, she silently added.
Irons nodded. "Don't take any unnecessary risks. Keep him talking as long as you can. We have teams searching his properties around the city, and every other likely location. With luck, we'll find the bomb before he can detonate it."
"Yes, ma'am. And if I could suggest something, maybe you should try to find his mother."
"Already in the works, Ms. Nash. But you're the one he wants to see. Good luck." Irons offered her hand, and Petra shook it.
All was silent, except for the click of her high heels on the pavement, as she walked the block to the Bartholemew Building. The light breeze felt cold on her exposed flesh, and she fought off a shudder. Uniformed Bugs and cops stared at her as she passed, and the high gravity of this world had never seemed higher. The entire planet pulled at her.
She entered the building and was shepherded to the elevator. A Bug she didn't know joined her on the ascent. "He has a little switch in his right hand," he said. "If he puts it down for any reason, just hit your wristcom's transmit button, and we'll be in the office a second later. But whatever you do, don't try to take the switch away from him."
They reached the floor of Bartholemew's office, and the Bug guided her out of the elevator, past dozens of tense security people. He pointed toward the door to the inner office and said, "We're all counting on you, Ms. Nash."
"Yes," she said. "I know."
Bartholemew was waiting for her, seated behind his big desk. In his neat, dark business suit, he didn't look much like an anarchist. He smiled at her, and Petra couldn't help returning it. She stopped a few feet in front of his desk and waited.
"So, you figured it out, did you, Petra Nash?"
"Isn't that what you wanted?" Petra asked him.
Bartholemew made a little head motion. "Perhaps," he said.
"I should have realized sooner. All that radical rhetoric. And dragging me out of the Old Annex just before you blew it up. And then insisting that we leave on the trip Sunday morning, just before the Emperor's speech. You couldn't have been more obvious if you'd left a trail of bread crumbs, but I was too dumb to see it."
"Not so dumb. I mean, here we are, aren't we? The fate of millions riding on our every word, perhaps the very future of the Empire itself hanging in the balance."
"I think the Empire will survive, no matter what happens."
"Your precious Empire will collapse of its own weight," Bartholemew said, letting his familiar anger show. "If not now, then later. I'm merely giving it a timely shove. Even if I don't get the Emperor himself—thanks to you—I'll still get my saintly old grandfather and half the bureaucratic offal in the Quadrant. People will see that there is nothing inevitable about Imperial rule."
"And they'll rise up and spontaneously overthrow their oppressors?" Petra asked.
"With a little help and guidance," Bartholemew replied. "Operatives from PAIN are on half the planets in the Quadrant, ready and fully capable of providing a revolutionary vanguard to lead the uprising."
"Maybe," Petra said, "but they'll have to do it without all those weapons you've got on GAC 4367. I found them, Whit. That's where Gloria is right now."
Bartholemew frowned. He seemed genuinely and unpleasantly surprised. Then he nodded and said, "That B & Q data?"
"That's right. It took some work, but eventually I figured out where that Savoy shipment had to have gone. And then I used the same process to track that freighter you leased in November. That's when you brought the bomb and the other weapons to New Cambridge."
"Right again," Bartholemew acknowledged. "That idiot Quincannon gave you everything you needed."
"And that's why you killed him?"
"One of the reasons," Bartholemew said.
Petra closed her eyes for a moment and saw, again, the battered corpse in the dark, old office. "You did it yourself?" she asked him.
"With my own two hands," he said, a note of satisfaction in his voice. "What's more, I personally set the grenades in the Old Annex building. I had some help for the others, of course, but I'm not one of those delegate-everything leaders. I wasn't afraid to get blood on my hands—literally, in the case of Quincannon. I did my share of the killing. A necessary overture to the symphony of destruction to follow. Anticipation is an essential element of terror, you see. Spreading fear and a sense of helplessness, underlining the authorities' impotence—it all contributes to the final result."
"More than two hundred people died that night, Whit," Petra said.
"And a hundred million more will die if—when—I flick this switch," he said, holding up the tiny device between his right thumb and forefinger.
"And you'll be one of them. So will I. So will your mother."
"An unfortunate but necessary sacrifice. We might have avoided that if you hadn't been such a dedicated little bureaucrat."
"Do you really want to kill them, Whit? Do you want to kill your mother? Do you want to kill me?"
"No," he said, "truthfully, I don't. Not you, not Mother, nor any of those faceless millions. Quincannon's another story. But I honestly have no desire to kill all those people."
"Then why do it?"
"Historical necessity. We anarchists understand that destruction is really the most profound act of creation. Did you know that a hundred million sperm cells die in order to fertilize a single egg? Today, a hundred million people will die to fertilize the egg of revolution."
"Bullshit!"
Bartholemew tilted his head to one side. "Yes," he said, "I suppose that one was a bit of a reach, wasn't it? They can't all be gems. But the point remains, and it is not merely bullshit. From the collision of opposing forces, new worlds are born. Better worlds. That is the inevitable result of the historical dialectic."
"If it's inevitable," Petra said, "then why do this? Why not let history work things out for itself?"
"Fabian heresy!" Bartholemew cried out in mock horror. "Like the benighted masses themselves, you lack the ideological underpinnings necessary to appreciate the beauty and necessity of revolutionary acts. History requires human agents to work its will. A few people among trillions understand that and have the courage and selflessness to make themselves into such agents. We offer ourselves as necessary sacrifices upon the altar of history."
"Oh, brother." Petra sighed. "Courage! Self-sacrifice! My goodness, I never knew you were such a great man, Whit. Here I thought you were just a bitter, resentful, angry, and confused guy who was pissed off at his father."
"Well...that, too," Bartholemew conceded, offering Petra a crooked smile. "Individuals have histories, no less than empires. I admit, if I'd had a happy home life, I probably wouldn't have spent twenty years working to build up PAIN, diverting funds, providing safe houses, and so on. It was the dialectic applied to the Bartholemew family, I suppose. The father works to build an empire, of sorts, and the son devotes himself to destroying it. Tell me, is that Hegelian, or merely Oedipal?"
"It's just sick, if you ask me. All this bullshit about historical forces—Spirit, Whit, do you really believe any of that stuff? Or is it just something you tell yourself to justify your fantasies?"
Bartholemew was silent a moment. Then he said, "Sometimes I believe it."
"And sometimes you don't?"
"Sometimes it's historical, sometimes it's personal. I admit the possibility that I'm wrong about the history. It's a big subject, after all, and it's possible that no one really understands it. But I believe that history requires us to act, in spite of our doubts and reservations."
"Did history require you to beat Jamie Quincannon to death with your bare hands?" Petra asked him. "He was a nice old man, Whit. I saw what you did to him."
"But you never saw what he did to me," Bartholemew said. His ruddy face darkened, and he looked down at his desk for a moment.
"What do you mean?"
"About forty years ago," he said, lifting his gaze to meet hers, "I was entrusted to the care of that nice old man. And he took me up to that office—the very same office where I killed him—and...and he did things. Hateful things."
"He molested you?"
"He did. Even then, I knew it was wrong, somehow. And I desperately wanted to tell someone. But my mother...somehow, I knew she wouldn't have understood. Or wouldn't have been able to do anything about it. I knew I needed to tell my father, but I couldn't. Quincannon was my father's partner, you see, and I was only his son. I didn't want to force him to choose between the two of us, because I knew what his choice would have been. So I never told anyone...until this very moment, Petra Nash. Since we are both about to die, it seems appropriate. Anyway, I grew up feeling as if _I_ had done something shameful and unforgivable. Eventually I realized how foolish I had been, but by then, it was too late to change anything. The child I was had grown to be the man I am."
"I'm sorry, Whit," Petra said. "That must have been awful for you. But do a hundred million people have to die just because you had a terrible childhood?"
"You think it's infantile revenge?"
"Isn't it?"
"If it were no more than that, you'd have a point." Bartholemew spent a moment staring at the device in his hand, then looked back at Petra. "I told you once that my father and I never really talked. But when he knew he was dying, he wrote me a letter that I received after his death. It was mostly just an exercise in self-justification, a litany of excuses for all his parental failures. Exactly the sort of thing you'd expect a man like that to say under the circumstances. But it also contained a few revelations that gave me a new resolve. You might say it was that letter that put this switch in my hand."
"What do you mean?"
"Dear old Dad tried to explain to me how it came to pass that he married my mother. It seems that my grandfather had made a big mistake. Details were not specified, but I gather that it was one of those grand, history-altering mistakes. Anyway, Norman Mingus, being intelligent, resourceful, and altogether unscrupulous, called upon Whitney Bartholemew to help him cover up that mistake. My father, you see, had the means available to help my grandfather dispose of the evidence."
"The Savoy shipment?"
"Precisely. Father's letter was unclear about exactly what happened and why, but he was explicit about the price he demanded in return for his help. The price was my mother, the lovely young Saffron Mingus. She was the belle of the Quadrant in those days, and she was engaged to Cornell DuBray, my grandfather's faithful assistant. But young Whitney Bartholemew had desired her from afar, and now he seized his opportunity to have her for his own. And my grandfather obliged him, gave him what he wanted. Again, details were lacking, but he promptly delivered my mother to my father as if she were...a shipment of arms."
Bartholemew smiled—to himself, it seemed to Petra. She realized that his words were for himself, as well. He had wanted her here merely as a sounding board, an audience for his final soliloquy.
"In the process of all this self-revelation," Bartholemew went on, "my father revealed to me the whereabouts of that Savoy shipment. He said that he had kept his part of the deal with my grandfather and had never attempted to move or sell those weapons. Mingus had apparently hoped that he would simply destroy them, but never really inquired. But old Bart was no fool, and knew that those weapons might give him considerable leverage if it ever became necessary to strike another deal with Mingus, who soon became Secretary of Dexta. I gather that he never had to use them for that purpose, but the potential was always there. Anyway, as a final gesture of filial affection, dear old Dad passed the secret on to me, his only begotten son. He figured that I could use that knowledge to extort Mingus for any favors that _I_ might need in the future. A wonderful gesture, don't you think? Except that Father never realized that I might think of another use for those weapons. I doubt that he even thought of them as weapons, per se, merely as potential blackmail material. But to me, they were precisely what they were intended to be—weapons that might be used against a powerful enemy. Like the greasy little criminal he was, Dad always thought too small. He thought he might use those weapons to blackmail Mingus and figuratively destroy him. I'll use them to destroy him physically, and much else. Perhaps even the Empire itself."
"And me," Petra added quietly. "And your mother."
"I'm truly sorry about that, Petra Nash."
"I don't want to die, Whit," she said. But she was certain now that she would. Her chin trembled and her eyes filled with tears.
"You're not going to cry, are you? I hate it when women do that. It's so terribly unfair."
"What were you going to do with me if I'd gone to Belairus? Keep me captive? Hold me hostage?"
Bartholemew shook his head. "I'd simply have loved you, as well and as truly as I could," he said. "I'd have run my various enterprises from there, of course, and continued helping PAIN as the revolution unfolded around us. But you would have been safe. And happy, I'd like to think."
"I don't think so," Petra said.
"Who can say?" Bartholemew asked. "But I wanted you to be happy, if that means anything. You made _me_ happy, at least. No one had ever really done that before. Oh, now, stop that crying! I told you, it isn't fair."
"But it's fair for you to kill me? Forget about the other hundred million people, Whit. You're killing _me_!"
"I'm sorry. But I no longer have any choice."
"Of course you do, Whit! The choice has _always_ been yours! Put it down, Whit! Put down your anger—and for Spirit's sake, put down that damned switch!"
"You heard her, dear. Put it down. I'm not ready to die yet, either."
Bartholemew looked up and Petra looked behind her. Saffron Mingus Bartholemew had entered the office, looking solemn and gravely beautiful. She stared at her son with loving, disappointed eyes.
"Put it down, Sonny," she said softly. "Put it down now. Make your mother happy."
Bartholemew looked at her, his eyes bright and shining. At last, he said, "Yes, Mother."
And he put it down.
GLORIA STOOD IN THE RECEIVING LINE AT THE entrance to the Ellisons' ballroom and watched as the Emperor approached. Black-clad Imperial Security forces were everywhere, but the mood was festive and relieved. The greatest threat to the Empire since the Fifth of October Plot had been overcome—at least, that was what the media said—and Gloria and the OSI had played a vital role in that triumph. It was a night to crow, and she felt proud and happy.
Charles, resplendent in his gaudiest Imperials, was accompanied by his cousin, Lord Brockinbrough, and Larry's son, Gareth, as well as a flock of aides, equerries, and ranking flotsam. First in the reception line was President Ogburn; he and Charles exchanged minutely calculated bows. The legal fiction maintained that the Emperor could not set foot on a planet without the invitation of the local government, and all the necessary rituals were observed.
Following Ogburn came the Ellisons and their son, hosts for the affair, then Quadrant Administrator Cornell DuBray, the Parliament Minister from New Cambridge, a collection of local and Dexta bigwigs, and finally, Gloria VanDeen, Director of the Office of Strategic Intervention, whose reception this was. Conspicuous by his absence was Dexta Secretary Norman Mingus. His daughter, Saffron Mingus Bartholemew, was also absent. And his grandson was in prison.
Gloria's satisfaction over the resolution of the PAIN threat was tempered by her awareness of the very personal pain it had caused Norman Mingus. Publicly, details about the Savoy shipment remained hidden, but it was impossible to hide the fact that Mingus's grandson had been deeply involved in terrorism. Unflattering facts about Whitney Bartholemew, Senior, had also been dredged up, and an unwelcome spotlight had been aimed at Saffron Mingus Bartholemew. The media were already speculating about whether Norman Mingus might be forced to resign over the affair.
Gloria could not bring herself to believe that it would come to that. Mingus had told her once that he would never resign, and, after forty-two years in office, he was far too wise in the ways of power to allow himself to be forced out.
The Emperor made his way down the line, seemingly relaxed and unhurried. Gloria marveled at Charles's ability to put commoners at ease and give them the impression that there was nothing more important in all the Empire than making small talk with his subjects. After two weeks of nearly nonstop schmoozing, Gloria thought she was beginning to get the hang of it, herself, and the experience would come in handy if she decided to become Empress. _If._
She still didn't know and hadn't had time to give the matter much thought. But the sudden reversal of OSI's fortunes made a future in Dexta seem more viable than it had a few weeks ago, when Erik Manko loomed large on her horizon. And, as Empress, would she be permitted the fun of dashing off to putrid scumworlds and risking her life in hand-to-hand combat with dangerous terrorists? Unlikely.
And then Charles was standing before her, handsome and grinning. Gloria gave him a little bow and let him clasp her hands. "A splendid reception, Ms. VanDeen," said the Emperor in a formal and audible voice. "We thank you for inviting us."
"And you honor OSI by your presence, Highness."
Formalities out of the way, Charles ran his eyes over Gloria's almost entirely naked body. She was wearing nothing but two strands of alternating diamonds and lapis lazuli, one low around her hips and the other hanging down from it to provide strategic, if symbolic, coverage, along with matching bracelets and earrings. She had last worn the gems on Mynjhino, on an evening that had ended in gunfire. No such excitement seemed likely tonight.
"Those diamonds look vaguely familiar," Charles said in a quieter, more private voice.
"They should," Gloria said. "You gave them to me for our second anniversary. You said the lapis lazuli matched my eyes."
"Ah, yes. I did, and it does. You look marvelous, as always, Glory."
Gloria leaned close to him and whispered in his ear. "You got my message?"
"I did."
"And you brought the...uh...item?"
"Right here in my pocket."
"Thanks, Chuckles. I owe you one."
"No, I think it is I who owe you. A nice piece of work, Glory, by all concerned. We'll talk later. For the moment, I still have some Imperializing to do." Charles kissed Gloria's hand, gazed into her eyes for a few long seconds, then moved on to mingle with the local gentry.
Larry and Gareth were next. Gloria shook hands with them and said, "Lord Brockinbrough, Gareth, thank you for coming tonight."
"The pleasure's all ours, Gloria," Larry said expansively. "I trust you'll be coming to visit my humble residence?"
"Not tonight, Larry," Gloria said. "Maybe tomorrow, after Charles's speech. I didn't realize that you had an estate here."
"One big one in each Quadrant, and a scattering of lesser hovels. My forebears liked to feel at home wherever they went. By the way, congratulations on your latest coup."
"Yeah," Gareth added, "that was pretty cool stuff, Gloria."
"Thank you, gentlemen."
The Brockinbroughs moved on, replaced by a Duke and Duchess, then a smattering of lesser Lords and Ladies. Gloria smiled her way through the rest of the formal presentations, then finally broke free to do some mingling of her own. She wandered out onto the dance floor and drank in the ambience.
Althea's little blues band had turned out to be a twenty-piece orchestra that specialized in twentieth-century music of all kinds. They played a few blues numbers, but also everything from Irving Berlin to the Beatles. When they struck up "In the Mood," Gloria found a partner and assayed an acceptable thirty-third-century version of the jitterbug. Three more men took turns cutting in on each other as they danced with her to the plaintive melody of "Yesterday." Then Gloria saw something that provoked a laugh and inspired her to do some cutting in of her own.
Elaine Murakami was dancing with Cornell DuBray to a Gershwin tune. Smiling, she tapped Elaine on the shoulder and said, "Pardon me, Elaine, but I'm pulling rank on you." Elaine giggled and got out of the way.
"Administrator DuBray," Gloria said, "I hope you're not planning to steal Elaine from me. OSI is very jealous of its personnel."
"Fear not, Ms. VanDeen," DuBray said. "Elaine is a delicious little treat in bed, but she wasn't much of a spy."
"You knew that I knew?"
"I figured it out quickly enough. You do have a way of inspiring loyalty in your people. In any event, I'll see to it that her father is released from prison."
"Thank you. You're very gracious in defeat."
DuBray raised an eyebrow. "I wasn't aware that I had been defeated."
"Well," Gloria said, "you certainly haven't won. In case you hadn't noticed, OSI is here to stay."
"Yes, I suppose it is. Very nicely played, I must admit. You won the hearts and minds of the Dexta masses, then capped it off with a dazzling bit of personal bravado. The Quad Admins can hardly dispatch you to bureaucratic limbo after such a performance."
"We're the Fifth Quadrant now," Gloria said. "Get used to it."
DuBray actually laughed at that. "As I told you, Gloria, we really are on the same side. OSI's triumph is Dexta's triumph, and Dexta's triumph is my own. However, we still have our differences, and don't imagine that the Quad Admins will simply bow before your brilliance. We won't, you know."
"I'd be disappointed if you did," she said. "I look forward to a long and lively rivalry."
"As a matter of fact," said DuBray, "so do I."
PETRA WISHED SHE WERE SOMEWHERE ELSE. Anywhere. Weehawken, even.
After what had happened the day before, she was not really in the mood for festivities and gaiety, least of all at the Ellisons' bemuraled mansion. Barely a month ago, she had enjoyed her triumphant entry into New Cambridge society in this same ballroom, but she could take no pleasure in the memory. She had lost Pug, she had lost Whit, and somehow, she felt that she had lost herself.
She didn't feel like being a Tiger tonight—maybe not on any night, ever again—but she had discovered that she had little choice. She'd had no time to run out and buy something conservative, so she found herself wearing a violet gown with a wide, deep neckline and a plunging back. Just the thing for a woman who romanced mass murderers.
On a night when she wanted to attract as little attention as possible, she found herself being asked to dance and offered drinks by a seemingly endless succession of smiling Dexta men. They offered their congratulations and expressed their admiration while staring at her daring cleavage, and Petra simply smiled grimly and tried to get through the evening without a complete emotional meltdown.
And there were the Ellisons to be endured. The haughty parents and their upward-bound son, with Steffany Fairchild at his side. She had exchanged formal and frosty greetings, then tried to avoid them. But she could feel them, staring down their noses at her, disapproving of her very existence.
She had also tried to avoid the media riot that followed the capture of Whit Bartholemew and the subsequent recovery of the missing plasma bomb. Dexta Internal Security had clamped a tight lid on the precise facts, but the Public Affairs Office could not resist exploiting such a triumph, and very much against her will, Petra had been trotted out at a press conference that afternoon. She had made a brief statement, then offered terse answers to a flurry of questions. When someone asked her about the nature of her relationship with Whitney Bartholemew, Junior, she had said only, "We were friends," then quickly exited the meeting.
Her friend had tried to blow up the city, and she had tried to put him in jail. Friends, indeed. They hadn't really been friends, at all. They had been hot, passionate, angry lovers, each seeking something indefinable from the other. Perhaps Whit had sought something normal in her, some link to an everyday existence that he despised; and perhaps she had yearned for something abnormal in him, an expression of defiance and rebellion against a world that had rejected her. Whit's father ignored him, but hers had walked out on her. Pug's parents pushed him upward; her mother seemed to want to pull her downward. One lover killed, another yanked away from her by his family, and a third who was nothing less than a monster. Some life.
There was a sudden regal fanfare from the orchestra, and the Emperor appeared on the bandstand. Petra watched from the far end of the ballroom and wondered what was going on.
"One of the nice things about being Emperor," Charles said as a hush fell over the crowd, "is that occasionally it is my lot to recognize and reward the accomplishments of certain of my subjects. While it is something I enjoy, it is not something I do lightly. Imperial honors are not easily earned, nor carelessly handed out. They signify that the person designated has served the Empire in a way that goes beyond the norm and is worthy of our highest recognition and gratitude. Tonight, it gives me great personal pleasure to honor one such individual. Would you please come forward and be recognized...Petra Nash!"
It took a moment to register. _Me?_
All around her, people were applauding. They opened a path for her, and Petra found herself walking dazedly forward, then up the steps to the bandstand, where the smiling Emperor awaited her. He put a hand on her shoulder and maneuvered her around till she stood next to him. Petra looked around, still not certain there hadn't been some mistake.
Charles raised an arm, quieted the crowd, then dipped his hand into a pocket of his tunic and withdrew a blue-and-gold ribbon with a large gold medallion dangling from it. "Petra Nash," he said, gazing directly into her eyes, "in recognition of your outstanding and meritorious service to the people of the Empire, and with deep personal gratitude for a job well and bravely done, I am pleased to present you, on behalf of three trillion grateful and admiring subjects, the Imperial Distinguished Service Medal." With that, he draped the ribbon around her neck. The golden ornament, with a profile of Hazar the Great etched on it, felt cold and heavy as it nestled between her breasts. She found it hard to breathe.
As the applause swelled, the Emperor bent down and said, "Congratulations, Ms. Nash. Would you do me the honor of the next dance?"
Petra gulped and nodded. Charles took her by the hand and led her down to the dance floor. The orchestra began to play "Moonlight Serenade" as the Emperor put his arm around her and guided her around the floor.
"I'm glad we finally had the chance to meet," Charles said. "Gloria often speaks of you."
"She mentions you, too, sometimes, Your Highness," Petra managed to reply.
"Favorably, I trust?"
"Uh...mostly, sire."
"Yes, well, I'm hoping I can enlist your aid in a most important matter, Ms. Nash. As you must be aware, I've asked her to marry me again and become Empress. I fear she's reluctant to leave Dexta, however, and I know she would not want to be separated from her dearest friend. So you must promise me that if Gloria agrees to become Empress, you will accompany her to Rio and serve as her personal assistant."
"Uh..."
"And, of course, if you do, you won't simply be Petra Nash."
"I won't?"
"No, you'll be Lady Petra of Weehawken. Sounds rather nice, doesn't it? You know, I don't believe I've ever ennobled anyone from New Jersey before."
Petra nearly tripped over her own feet, but the Emperor smoothly rescued her and smiled down at her. At six feet four inches, he towered over her.
"Your...Your...uh, Highness," Petra stammered, "I...I..."
"You will help me persuade Gloria, won't you? As Emperor, I know the importance of having strategic allies."
"I...uh..."
Charles laughed indulgently. "Say no more, Ms. Nash. Just keep my request in mind, if you would."
"I certainly will, Your Highness."
"Splendid." As the song ended with the familiar swirl of Miller reeds and muted horns, the Emperor leaned over and gave Petra a kiss on her lips. Then he stepped back and led the crowd in applauding her once again. Amid the kaleidoscope of color and noise, Petra saw the Ellisons standing at one side of the room. Mr. and Mrs. Ellison were clapping politely, Steffany Fairchild looked supremely miffed, but Pug was staring right at her, grinning, and slapping his hands together with enthusiasm. Petra grinned back at him.
Maybe she was glad to be here, after all, she thought.
AMONG HIS OTHER ACCOMPLISHMENTS, ELI Opatnu proved to be an excellent dancer. To the strains of "I've Got You Under My Skin," he whirled Gloria around the dance floor with grace and aplomb.
"I'm so happy for Petra," she told him. "She's had a tough time of it lately."
"Honors well deserved," Opatnu agreed. "And you didn't do so badly, yourself. But then, you already got a medal for Mynjhino, didn't you?"
"You'll get one of your own someday, Eli."
"I doubt it," he said. "Certainly, I won't get one for what I have to do now. Gloria, you have to shut down the investigation of Wendover and the double-flagging operation."
"What do you mean, I _have_ to?" she demanded angrily. "Where do you get off telling me to do a thing like that? I realize it might be an embarrassment for your Sector, but—"
"You don't understand, Gloria. I'm not telling you this for myself. I'm speaking on behalf of Ed Smith."
Gloria looked up at Opatnu in openmouthed shock. In return, he gave her a guilty shrug.
"Did you imagine you were the only one in Dexta who had a debt to the zamitat?"
"Spirit!" Gloria breathed.
"I'm just thirty-seven," Opatnu said, "and I'm already a Level Seven and a Sector Administrator. I'd like to think that my native abilities had something to do with that, but the truth is, I've had some help. Of course, that kind of help isn't free. You should realize that by now, Gloria."
Gloria took a deep breath and let the air out very slowly. "I guess I did realize it," she said. "I just didn't think I'd be making a payment so soon. It won't be easy to tell Jill."
"Jill will get over it," Opatnu assured her. "And what does it matter? This Wendover thing is just routine. The zamies aren't asking you to sell your soul to Beelzebub. They just want a little help. They turn a few extra crowns on the double-flagging, a couple of minor Dexta officials get some kickbacks, and the Empire loses a little tax money that it will never miss. As I said, it's routine."
"And what's in it for you, Eli?"
"The continuing gratitude and cooperation of Ed Smith and people like him. I get to look good, appear to be clean, and go on doing what I think has been pretty good work for the people of the Empire. The same goes for you, Gloria."
"I suppose so," she said. Ever since that meeting in the restaurant with Ed Smith, she had known this day would come. She had accepted it then, and she had to accept it now. It was just quid pro quo, after all. Routine.
OPATNU CAUGHT DUBRAY'S EYE, THEN FOLLOWED him into an unoccupied sitting room just off the ballroom. "You told her?" DuBray asked him.
Opatnu nodded. "The deed is done," he said.
"And how did she take it?"
"She accepted the necessity. You know, this whole thing would have been much easier if you'd told me in the beginning that she has a debt to our friends."
DuBray shrugged. "If I had known, I would have. That's one of the problems in dealing with our friends. Half the time, one of their hands doesn't know what the other hand is up to. Fortunately, I got to wondering how she dealt with Manko and made some inquiries. It surprised me a bit, I can tell you. Pure and high-minded Gloria VanDeen!"
"She's as human as the rest of us."
"And as flawed, it would seem. In any case, our friends will be pleased that the investigation has been buried. With the new product ready to hit market, this was no time for complications."
"I just wonder how Jill will take it," Opatnu said.
"Jill?"
"Clymer. The one who was running the investigation."
"Of course. Why, will she be a problem?"
"Probably not. But she won't be happy about it."
DuBray clapped a hand on Opatnu's shoulder. "Take some advice, Eli," he said. "Stop worrying about other people's happiness. We don't do these things to be liked, you know."
"Why do we do them?"
The two men stared into each other's eyes for a long, silent moment. Finally, DuBray said, "What else is there?"
"WHY DO I HAVE TO DISCONTINUE THE INVESTIGATION?" Jill Clymer demanded. "I've worked hard on this, Gloria, and I don't see why we should just stop in midstream."
Jill had raised her voice a little when Gloria gave her the news, so Gloria took her by the arm and led her off to one side of the ballroom. The band was playing "With a Little Help from My Friends."
"Jill," Gloria said, "I appreciate all the work you've done. But the way things are shaping up, with the restraining order and everything, it looks as if this could drag on for months or years. I just don't think this is the kind of thing that OSI should be getting involved with. Send me all the files you've put together, and I'll see to it that the Comptroller's Office gets them. It's more their sort of thing, and maybe they'll want to pursue it. But as of now, OSI is out of it."
"Is this your decision," Jill asked, "or are you getting pressure from higher up?"
"Nothing from higher up," Gloria replied, accurately, if not with complete honesty. "It's my decision."
"Well, I don't like it."
"You don't have to like it, Jill. You just have to do it."
"I see." Jill stared at her for another moment, then turned and walked away.
It hadn't been so difficult, after all. In fact, Gloria was mildly surprised by how easy it was.
Routine.
HAPPY BUT EXHAUSTED, GLORIA AND PETRA returned to the suite at the Imperial Cantabragian sometime after two in the morning. During the limo ride, Petra had alternately stared off into space with a distracted half smile on her face and babbled almost incoherently. The prospect of becoming Lady Petra of Weehawken had all but unhinged her. At the same time, she was well aware that Gloria had not made up her mind about becoming Empress.
"Maybe you could do both," Petra had suggested. "Be Empress Tuesdays, Thursdays, and weekends, and Director of OSI Mondays, Wednesdays, and Fridays."
"I don't think that would work very well."
"Then alternate weeks? Or months?"
"You really want that title, don't you?"
"Well...yeah, I guess so. Spirit, Gloria, I don't know. I mean, I'd hate to leave Dexta, but...well, you know..."
"Yes," Gloria said, "I do."
"And the Emperor!" Petra gushed. "I mean, he's just so handsome and tall and dreamy. How can you resist him, Gloria?"
"I've had plenty of practice."
"Oh. Yeah, I see what you mean." Petra had lapsed into silence for a while and stared out the window of the limo. Then a goofy smile crept over her face and she said, "I danced with the Emperor!"
They walked into the suite, with Petra humming "Moonlight Serenade," and Gloria noticed the message light on the pad she had left on the coffee table. It proved to be a message from Norman Mingus, asking her to come upstairs to see him when she got in, no matter how late.
After a moment's indecision, Gloria reluctantly removed her diamonds and slipped into some jeans and a tee shirt. She knew Mingus liked to see her in full flower, but she had a feeling that this was not the right time. She said good night to Petra and went to the elevator.
There was still a gaggle of Bugs upstairs, but she breezed through them, was met by an aide, and directed into Mingus's bedroom. She found him there, in pajamas, robe, and slippers, sitting quietly in a comfortable chair. On the table next to him stood a bottle of what appeared to be Belgravian whisky, reputedly the best in the Empire. No milk and cookies on this night. Gloria walked over to him, gave him a kiss on the cheek, then sat down in a chair next to his.
"You missed a good party, Norman," she said. "I'm sorry you weren't there, but I understand."
"I saw some of it on the vid," he said. "You looked very enticing. And I'm pleased for Ms. Nash. She deserved that medal."
"And she gets to be Lady Petra of Weehawken if I go back to Charles," Gloria said. "But I think it would be nice if she also got something from Dexta."
"Yes," said Mingus, "I had thoughts along the same line. She's a Thirteen now? Very well, in the morning, you can tell her she's a Twelve."
"Thank you, Norman. That's very sweet of you."
"Nonsense. She's earned it." Mingus turned a little in his chair to look directly at Gloria. "As for you, young lady, I ought to demote you for that harebrained stunt."
"You gave me the job, Norman," Gloria said placidly, "and harebrained stunts were always a part of it."
"I never meant for you to put yourself at risk like that."
"If I hadn't been there, it would have turned into a stalemate, or worse. I did what I thought was necessary, and it worked."
"I can't deny that," said Mingus. "Nevertheless, I would prefer it if your personal interventions remained strategic rather than tactical."
"Don't get too attached to me Norman," Gloria advised. "You could lose me, one way or another."
Mingus stared at her for a moment. "Charles?"
"I'm still considering it," she said. "I'm going to spend some time with him tomorrow after his speech. Maybe I'll be able to make up my mind."
"I see," he said. "Naturally, Gloria, I hope that you'll decide to remain with Dexta. But you must do what you think is best for yourself. Forgive me, would you care for some whisky? Or anything else?"
Gloria shook her head. "No thank you. I'm fine."
Mingus lifted a tumbler to his lips and took a slow sip of the Belgravian whisky. He held the glass there for a moment, wordlessly staring off into the mid distance.
"Norman? I'm awfully sorry about what's happened."
"As am I," he said as he put the glass back down on the tabletop. "Oh, I'm certain I shall survive the storm, personally. Internal Security has seen to it that those who are aware of what transpired between Ms. Nash and my grandson will keep their mouths shut. What scandal there is will quickly dissipate. My grandson's trial will be a private affair, under Imperial Security, and no one will ever learn the full truth of the matter. Except for you, Gloria. That's why I asked you here tonight. I think you deserve to know."
"Petra told me what Whit said, so I think I have some idea of what actually happened."
Mingus shook his head. "No," he said, "you don't. Only four people ever knew the whole truth of it, and now one of them is dead. Cornell DuBray and my daughter are the others."
"You don't need to tell me, Norman," Gloria said.
"Yes," he said, "I do. I need to tell it, and you need to hear it. If you truly intend to run Dexta someday, you should know what that entails. You should know what it might cost you."
Mingus poured some more whisky, then took a healthy swig of it. "Belgravian," he said. "Are you sure you won't have some?"
"Maybe a little." Mingus reached for a second tumbler and poured a couple of fingers of the amber liquid into it. Gloria took a sip of it and let its silky smoothness caress her tongue for a moment.
"You know that I was Quadrant Administrator here in the summer of 3163," Mingus began. "I already had prospects of rising to the position of Secretary, but that was not yet a certainty. My predecessor, Tom McIntyre, still had a few good years left in him, and I had a potential rival or two. Still, my prospects were good, and I intended to make the most of them. I tell you this because it's important that you keep in mind as you hear what follows the central fact of my personal ambition. Whatever my other motives and justifications for what happened, they cannot be separated from my sense of self-interest, my sense of entitlement. If you are to judge me—and you will—you must keep that in mind."
"I have no intention of judging you, Norman," Gloria protested. "I don't want to do that."
"You must," he said. "And inescapably, you will."
Mingus drank some more whisky and focused his gaze on the darkness in a far corner of the room. "We were going to have a war," he said. "A big one. Everyone knew it. We had been fencing with the Ch'gnth for hundreds of years, and by the middle of the last century, conflict had become all but inevitable. The Emperor, Edward III, had been on the throne for more than thirty years and had just a few more years to live. He was not a bad Emperor, and he had some competent people around him. And Tom McIntyre was, at his best, quite capable. Yet there was a pervading sense of drift in those final years before the war, a lack of focus. We were not truly prepared for the war we knew was coming. I suppose no one quite believed that it would really happen. The Empire hadn't fought a major war in nearly a century, and the prospect of one seemed unreal, somehow. We stood on the brink of a precipice, idly playing lawn croquet as the ground crumbled beneath our feet, and wondered only how that would affect our next shot. It's one of those things that historians and posterity can never truly understand because they weren't there and didn't breathe the same air as we did.
"So there I was, Administrator of a Quadrant that was about to explode. And I knew, to a near certainty, that when the Ch'gnth attack came, it would be focused on Savoy, just seventy-five light-years from where we sit. Just three days away for a battle fleet. Ch'gnth space was closed to us, of course, but we had some scattered intelligence reports from neutral traders, and we knew that the Ch'gnth were assembling a powerful strike force in the region. And, of course, galactography virtually dictated that they would attack Savoy. It was a salient, intruding into space that was rightfully theirs. Once they had it, the entire Empire position in this Quadrant would be at risk. Their next target would surely have been New Cambridge, and from there, they would have had a clear path leading to Earth itself. Thus, the strategic position."
Mingus took some more whisky, then continued.
"We weren't blind to it, of course. I had been sending warnings to Earth for years. I was rather strident about it, in fact, and I suppose that contributed to their tendency to discount my recommendations. In any event, little was done to strengthen materially our position here. Oh, there were plans aplenty, the occasional Fleet Exercise and what have you, but somehow the essential gravity of the situation never truly penetrated. It wasn't until negotiations finally broke down in June of 3163 that anyone in authority on Earth fully appreciated the urgency of the situation. So they quickly threw together a shipment of arms and sent it to New Cambridge, for transshipment to Savoy. It arrived here in August. But by then, it was too late." Mingus looked at Gloria and added, "Or so I believed."
"The history books say that you were the only leader at the time with the vision to see what was coming," Gloria said.
Mingus allowed himself a marginal smile. "The history books," he said, "are wrong. As usual. There were others. Even as I sat here dithering on New Cambridge, Admiral Bryant was already pulling together the pieces of what would become the Second Fleet. That's important for you to understand. For all my presumed vision and foresight, I was only viewing things from my own limited perspective. Here in Quadrant 4, things looked truly desperate. But there were three other Quadrants, you see. The Empire was never quite as fragile and vulnerable as it appeared to me. Others, with a broader vision—like Admiral Bryant—had a more comprehensive and realistic understanding of our position. But I could see no farther than Savoy. And what I saw was that Savoy was doomed."
Mingus shook his head sadly and sighed. He took another sip of whisky.
"I couldn't see it then, of course, but from the distance of half a century, it's clear to me now that in August of 3163, I was in an advanced state of panic. I envisioned the swift and final destruction of Savoy, followed by an inexorable, irresistible attack on New Cambridge. In my mind, Savoy was already lost. Gone. All that mattered was to preserve New Cambridge. Savoy could not be saved, but New Cambridge might be, if we could husband what strength we had and do what was necessary for our defense."
Mingus looked at Gloria. His blue-gray eyes were misted and shining.
"And so," he said, "I didn't send that final shipment of arms on to Savoy. I kept it here, for the defense of New Cambridge. I knowingly decided to sacrifice a hundred million lives on Savoy in order to save my own skin."
"No," Gloria insisted. "You're being too hard on yourself. You already said that you didn't think Savoy could be saved, no matter what you did. Given what you knew and believed at the time, you did the right thing. The only thing possible."
"Wouldn't it be nice to think so?" Mingus said. He poured himself some more whisky. He took another sip and so did Gloria.
"Norman, you can't—"
Mingus raised his hand. "Hear me out, Gloria."
"All right, but I refuse to believe you panicked."
"Call it what you will. If you prefer, say simply that I tragically misjudged the situation. For when the Ch'gnth attack came, on September 8, those weapons were stored in an orbital warehouse above New Cambridge, when they should have been deployed on and around Savoy, where they might have done some good. We heard about the attack from a courier that got through a day later. I fully expected that Savoy would fall within a few days, a week at most. And if it had, I suppose my decision would have been fully justified. That single shipment of arms would not have changed the outcome of the battle. The Ch'gnth force was overwhelming, and would surely have prevailed in time. But time"—he shook his head sadly—"time was what mattered above all.
"Savoy did not fall within a few days. Nor within a week, or even two. It held out for nearly three weeks. Three desperate, bloody, heroic weeks." Mingus shook his head again. For a moment he seemed too overcome with emotion to continue. Gloria made a point of looking away as Mingus dabbed at his eyes.
"Afterward," Mingus went on, "we were able to reconstruct what had happened. To truly understand it, you need to know something about the tactics and weaponry that were employed at the time. In the initial wave of the attack, the Ch'gnth engaged and defeated Savoy's orbital defenses. That was inevitable and nothing could have prevented it. They were just too strong, and Savoy was too weak—in space. But on the surface of the planet, Savoy possessed formidable defenses—and a will to fight that, even now, makes the hairs on the back of my neck stand on end when I think of it."
"Mine too," said Gloria. "I remember learning about it in school when I was a little girl. My teacher cried when she told us about the defense of Savoy. We all did."
Mingus cleared his throat. "Savoy," he said, "has a single major continent, and most of the population was concentrated around Savoy City, on the northern coast. The city was impervious to attack from orbit. The point defenses against space-borne or ballistic projectiles were simply too strong. The only way to get at the city was from the surface. The task facing the Ch'gnth was to establish a bridgehead on the surface, then launch terrain-following missiles aimed, ultimately, at the city. They made three landings. The first was repulsed. So was the second, after a pitched battle that lasted six days. But the third landing succeeded, and the Ch'gnth established a defensible perimeter in a weakly defended desert region, about two thousand kilometers southeast of the city.
"From there, they were able to unleash their terrain-following missiles and literally blast their way forward, toward the city. I saw it from orbit a few months later, and you could see each crater, each blast zone, marching straight as an arrow aimed at the heart of the city. They'd launch a missile and detonate a plasma bomb perhaps twenty kilometers beyond their lines. There simply wasn't time to defend against such an attack; by the time the defenders spotted the launch, the warhead had already exploded. One after another, day after day, more than a hundred of them. Until finally, nineteen days after their initial attack, they reached the city. And annihilated it."
Mingus closed his eyes, leaned back in his chair, and took a deep, slow breath. Then he went on. "By then, of course, they had already exterminated the outlying settlements. And on the nineteenth day, they killed every remaining human being on the planet."
"A hundred million of them," Gloria said softly.
"One hundred and three million, two hundred and seventy-nine thousand, four hundred and ninety-one," Mingus said, "according to the 3160 census. Of course, by September of 3163, it would have been more. But for a round figure, I suppose one hundred million will do."
Mingus poured himself some more whisky and took a big swallow. Gloria sipped some more of her own drink.
"Nineteen days to kill a planet," Mingus said. "And on the twenty-first day, Admiral Bryant and the Second Fleet popped out of Yao Space, suicidally close to the planet, and blasted the Ch'gnth fleet to perdition. They speak of Salamis and Lepanto, Midway and Caliban Four, but really, there had never been anything quite like it. If the navigation had been off by a thousandth of one percent, they'd have smashed into the planet or missed it entirely. If the timing had been off by a tenth of a second, they could not have hit the Ch'gnth fleet. The Spirit must have been with us." Mingus shook his head.
He looked at Gloria. Tears were running down his cheeks. "Do you see now what I had done?" he asked her. "Do you understand the enormity of my crime?"
"Norman—"
_"Two days!"_ Mingus shouted. "Two Spirit-forsaken days! That last shipment of arms, which I, in my vast wisdom, withheld because Savoy was already doomed and could not have been saved—can you tell me that it would not have made a difference? Can you tell me that it would not have given those poor, brave souls the time they needed? A hundred million people, who might have lived, died because of what I did! No, that's wrong. A hundred and three million, two hundred and seventy-nine thousand, four hundred and ninety-one. I mustn't shortchange them. They all counted. Every man, woman, and child. They all counted."
Mingus put a hand over his face and sat there, quietly sobbing. Gloria wiped tears from her own cheeks.
"Norman," she said, reaching for his hand, "you couldn't have known."
"Well, I damned well _should_ have known, shouldn't I?" Mingus took a handkerchief out of the pocket of his robe and wiped his face. He cleared his throat and helped himself to more whisky.
"Anyway," he said, "the outcome of the Battle of Savoy left me in a rather embarrassing position. The weapons that might have saved Savoy, the weapons I had withheld, were sitting up there in those orbital warehouses. What was I to do with them? If what I had done ever came to light, I'd have been pilloried...lynched. As I should have been. I couldn't just hand them over to Admiral Bryant and say, 'Here, old boy, you might need these,' now, could I? The survival of Savoy had become a moot issue, and all that mattered was the survival of Norman Mingus."
Mingus drank some more whisky. Gloria thought he was getting a little drunk; his words were slightly slurred and his face looked unnaturally red and puffy.
"If I was to avoid disgrace, or worse, I had to dispose of those weapons. Fortunately, I happened to know a man who had the means to do that. His name was Whitney Bartholemew, and he was the neighborhood distributor for organized crime. The zamitat. I'd had dealings with him, of course—it's unavoidable."
"I know what you mean," Gloria said quietly. Mingus ignored her.
"So I approached him and explained my situation. He was not unsympathetic, and offered to help me in my hour of need. I asked him what he desired in return. Money? Official protection? His for the asking! Only he didn't ask for that. He wanted only one small, inconsequential thing in return for his assistance. He wanted my daughter."
Mingus turned to Gloria and offered her a self-deprecating smile. "Poetic, don't you think? Downright Shakespearean. I was like Shylock, crying, 'Oh, my daughter! Oh, my ducats!' Only I was the borrower, not the lender, and life was extracting its pound of flesh from my very heart. I was properly shocked and offended, of course, like the good father I pretended to be. But I knew as soon as he asked what my answer would be.
"Saffron was betrothed, at the time, to my friend and assistant, Cornell DuBray. I knew that Saffron would never understand, but that DuBray would, so I went to him. And like the master politician and bureaucrat I was, I proposed a plan that I knew would be acceptable to him. In return for my protection and patronage throughout his career at Dexta, he would break off his engagement to my daughter. And it would be done in such a way—as finely calculated as Admiral Bryant's attack at Savoy—that Saffron would, with my subtle encouragement, be thrown directly into the waiting arms of Whitney Bartholemew. And that is precisely the way it happened. A masterpiece of creative plotting, if I do say so myself—some of my finest work. Saffron was never to know, of course. And in the meantime, Bartholemew disposed of the weapons. I never knew just how or where until you and Ms. Nash unraveled the mystery this past week."
Mingus allowed himself some more whisky. "All very neat and tidy, you must admit. Except for one thing. After they were married, for some petty reason or another, Bartholemew told Saffron what had happened. You can imagine how she reacted. No, I suppose you don't have to imagine it. You already know. She hated me. And, whether intentionally or not, she passed that hatred on to her son, my grandson...with the tragic results you have seen. So you can add a few more to my score of one hundred and three million, two hundred and seventy-nine thousand, four hundred and ninety-one blighted lives—another two hundred from the terrorist attacks on Central. Plus two more."
Gloria squeezed his hand. She badly wanted to help him, to ease his pain, but she could think of nothing to say.
"I tried to speak to Saffron after my grandson was arrested," Mingus said. "But she wouldn't speak to me. She probably never will again. As for Whitney, there is little I can do for him. It's an Imperial matter, and the outcome of his trial is a foregone conclusion. They'll execute him for what he did and tried to do. Perhaps, in his perversity, that will even make him happy. He'll get to be a martyr."
Mingus sniffed and wiped away some more tears. "There is, perhaps, one thing I could do for him, and I briefly considered doing it. I could make a clean breast of it. Confess. Tell the Empire the truth about Savoy and my crimes."
"Norman, you can't do that!"
"Of course not. And I won't. Just an idle fantasy. No, I shall continue as before, but with the additional burden of knowing the irreparable harm I have done to my own family."
He looked into Gloria's eyes. "For fifty-four-and-a-half years, Gloria, not a single day has gone by when I have not thought of Savoy, when I have not felt the pain and guilt and remorse. I shall feel them to the end of my days."
"I'm so sorry, Norman," she whispered.
"Do you still want to run Dexta?" Mingus asked her. "Spirit willing, nothing like Savoy will ever happen to you. But if you truly seek that power, then it is all but guaranteed that something else will happen. Something uniquely yours, heartbreaking and inescapable. Having power means making decisions, Gloria, and because you are human, some of those decisions will be wrong—perhaps fatally so. And you will have to live with the consequences, with the responsibility for whatever tragedies are unleashed by your mistakes and frailties. With the best of intentions, I condemned a hundred million people, and because I was clever and selfish, I managed to evade the direct consequences of my actions. They fell, instead, on my family, and on total strangers. And yet, I know. _I know._ And you will, too."
"Oh, Norman," Gloria said softly.
Mingus managed to smile at her. "Thank you for listening to an old man's lachrymose confession, Gloria. I needed to tell someone. I'm sorry to have burdened you with this knowledge, and yet, perhaps it will help you to make the decision that you face. I love you, Gloria...like a daughter. And if you truly were my daughter, I think I would tell you to forget about power and responsibility and simply to live a happy and carefree life. Go be a smiling, glamorous Empress, and avoid the kind of pain I have known. That is what I would wish for you."
"You want me to leave Dexta?"
"I want you to be happy. And now, I find that I am very tired. Sweet dreams, Gloria."
"And to you, Norman."
He shook his head. "My dreams are never sweet," he said.
THE EMPEROR'S PERSONAL LASS ROSE FROM the grounds of the soccer stadium and into the air above Central. Gloria looked out of the windows, down at the sprawling city, and wondered what it would have looked like if Petra hadn't saved it. It would have looked, she realized, like Savoy.
Charles's speech had closed out, at long last, the Quadrant Meeting. Bureaucrats by the hundreds would soon be crowding the port and Orbital Station, bound for their distant domains, where they would carry on the endless work of empire. And the ordinary people of New Cambridge would get on with their lives, secure in the embrace of a government that had protected them from the unsleeping evils of the galaxy. Under a handsome young Emperor who loved his subjects.
"I hate making speeches like that," Charles said. "I don't mind the smaller, formal affairs so much, but these big, outdoor extravaganzas give me delusions of grandeur. I feel like Hitler at Nuremberg or Hazar at Golconda. I raise my arm and thousands cheer. I clear my throat, and thousands cheer. I could fart, and thousands would cheer."
"As long as it's cheers and not jeers," his cousin Larry said, "what are you complaining about?"
"Point well-taken," said the Emperor. "Still, the unreality and absurdity of it all bothers me. The Caesars used to have slaves who stood behind them, whispering, 'Thou art mortal,' or some such thing, just to keep their feet on the ground." He turned to look at Gloria and added, "But I'll have you to do that for me, won't I?"
Gloria smiled at him. "Maybe," she said.
"You've looked over the agreement my people prepared?"
"Glanced at it," Gloria said. "It looks pretty good, but I have some questions about a few of the specifics. We'll talk about it later."
"As you wish." Charles gave her a probing stare. "Is something wrong?" he asked.
Gloria shook her head. "I'm just tired," she said. "And I have a lot on my mind."
"I'll not press you, then." Charles went forward to chat with the pilot, while Gloria gazed out the window at the glowering cliff face and the churning waters below.
She had not slept at all last night following her meeting with Mingus. It had drained her emotionally and left her feeling adrift. To see Norman Mingus sobbing in grief and remorse was almost more than she could handle. And his final words of advice confused and unsettled her.
Even though the agreement Charles had offered promised her real power, she knew that the accomplishments of an Empress could never measure up to what she might do—had already done—at Dexta. The real power, the life-and-death power, would remain with Charles, as it had to. The weight of responsibility would fall only lightly on the shoulders of an Empress, and perhaps she might be vouchsafed the happy and carefree life that Mingus wished for her.
Poor Norman! She ached for him and wished that there were something she could do for him. She wondered how he had endured so much pain, for so long. Could she endure such a burden, or would it crush her?
Back on Mynjhino, less than two years earlier, she had saved, perhaps, millions of lives. The pride and satisfaction she felt were beyond description. Yet, how would she have felt if it had gone the other way? What if she had made some fateful, fatal mistake, and instead of saving millions, had killed them? Could she have lived with the knowledge, the guilt, the way Mingus had?
It hadn't happened that way. But someday, it might. According to Mingus, it almost certainly would, because she was human, and because Dexta offered the kind of power that few humans had ever possessed. Perhaps humans _shouldn't_ have that much power. Maybe they should just turn the whole thing over to computers. The computers would probably make mistakes, too, but would they feel pain and guilt over it?
The anguish of Norman Mingus frightened Gloria. If a strong and good man like Mingus could be so tragically wrong, and so haunted by it, who was she to think she could do better, or as well? Did she really even want to try?
Dexta! Empress! She smiled to herself and thought, _Do I want to be a Lady or a Tiger? Which door should I open, which should I close?_
THE BROCKINBROUGH ESTATE SPRAWLED ALONG the cliff top north of Central, overlooking the narrow straits. The main building looked like a medium-sized museum, with columns and arches and domes, while the many outbuildings made Gloria think of a feudal village surrounding a castle. The LASS settled onto a vast greensward in front of the main building, and the occupants debarked amid much bowing, scraping, and fussing by the Brockinbroughs' liveried attendants. Black-clad Imperial Security personnel were much in evidence, but they didn't seem to have a lot to do. The Emperor's arrival was routine in every respect.
Larry and Gareth led Charles and Gloria into the building, playing at being tour guides. Larry explained that the original building had been constructed five hundred years earlier, with various new additions and appendages added as the centuries passed. Some artwork and sculptures by renowned artists were worthy of particular note, especially a life-size rendering of the Spirit that had been done in the early 2900s by Komari. "The Imperial Museum in Central has been begging us to let them have it," Larry said. "Good thing we didn't."
Charles nodded distractedly. Larry noticed the Emperor's lack of interest.
"Perhaps we should save the grand tour for another time," Larry said. "I can see that you and Gloria have other things in mind."
"Can't slip anything by you, can we?" said Charles, who had his arm around Gloria's waist.
"Allow me to conduct you to your quarters, then." It took nearly five minutes to reach them, but it was worth the wait. The bedroom was huge, with high, ornate windows overlooking the strait, and furnishings that would have done Louis XIV proud. The adjoining bathroom gleamed with late-Roman-era decadence.
"And then," Larry said, conducting them through another door, "there's this." He ushered them into a smallish room that was covered—floor, walls, and ceiling—in a deep blue plush fabric that was soft and bouncy. There were no windows or furniture, and only the single door. "A null-room built for two," Larry explained. "Enjoy. Gareth, let us leave our distinguished guests to their own devices—I'm sure they'll manage without us."
"The null controls are voice-activated," Gareth said as he and Larry left the room and closed the door behind them.
Charles turned to Gloria. "There, now," he said. "And you always say that Larry is a thoughtless and selfish jerk."
"He's probably got peepholes hidden somewhere around here," Gloria said.
"Then we should give him something to peep _at,_ shouldn't we?" Charles pulled her close and pressed his lips against hers. When he began pulling at the thin fabric of her dress, Gloria backed away.
"One second," she said. She opened her handbag and spoke into her pad. "No messages or calls for the next three—make that four—hours. No exceptions." Then she put the bag down on the soft floor and looked at Charles. "I'm all yours, Chuckles."
"For four whole hours. I'd much prefer to measure our time together in years, Glory. Or decades."
"We're both too young to understand what decades mean," Gloria said, thinking of Mingus's five decades of anguish. "Maybe we should just concentrate on the present."
"I'm willing," Charles said. "But I need an answer, Glory. If not now, then soon."
"I know," she said. "And I'm sorry it's taken me so long. It's just that there's so much to think about." She grinned at him. "For what it's worth, I think you got Petra's vote last night."
"Charming young woman," he said. "I can see why you like her so much. But if I have the Petra Precinct locked up, perhaps I should do some more campaigning in _your_ district."
"What did you have in mind? An Imperial Poll?"
"Interest is already rising," Charles said, "and I predict an imminent eruption of popular support. I intend to stuff your ballot box, my dear."
"Vote early and often," Gloria urged him.
Their garments were soon scattered around the room. They wrestled and writhed on the soft floor for a few minutes. Then Charles said, "Computer. We'd like to float about four or five feet off the floor."
There was a sudden soft hum, coming from all around them, and they clutched each other's bodies as the null field engaged and gently lifted them into the air. The field stabilized, and Gloria felt the soft pressure from above and below, caressing her like a warm bath. Charles skillfully orbited her, his fingers, tongue, and lips precisely mapping her prominences, bays, and declivities until, his circumnavigation complete, he made his final approach and accomplished his landing with confidence and finesse. They spun and twirled together, locked in a celestial embrace, like ancient gods or randy avian creatures. Gloria raked her fingernails across his back, inspiring redoubled Imperial ardor, until, at last, Charles grunted, growled, and gasped in the ecstasy of release, and Gloria followed him a moment later.
They clung to each other as the tide receded, content to float on the aimless currents like driftwood after a hurricane. Gloria sighed happily and wondered why it couldn't always be like this with Charles. Maybe, she thought, it could. Maybe...
She abruptly became aware that the door had opened, and Larry and Gareth were standing there, staring and grinning at them. Charles raised his head and snarled, "Just what the hell do you think you're doing, Larry?"
"What do I think I'm doing, good cousin?" Larry asked. "Why, I think I'm assassinating you!"
Before Charles or Gloria could respond, Larry manipulated some controls on a small electronic device in his hands, and suddenly the air pressed down on them. They fell to the floor with a heavy, jarring impact, but the pressure from above never relented. Gloria felt it crushing her downward against the soft blue of the floor, squeezing the very air out of her lungs. She fought against it, but she was immobilized by the tyrannical, invisible weight. Beside her, she could sense but not see Charles fighting his own futile battle against the enveloping force. She gasped for air but could find none.
"You're smothering them, Dad," said Gareth. "Here, let me." A moment later, Gloria felt the pressure lessening around her head, and she was able to open her mouth and gulp a hard-won breath. But the pressure on her chest and limbs never relented.
"Sorry," said Larry. "Gareth is much more adept with these gizmos than I am." Gloria managed to tilt her head upward enough to see that Gareth now had the control device in his own hands. Father and son continued standing just beyond their feet, smiling at them.
"Are you out of your mind?" Charles demanded in a labored gasp.
"Not in the least," Larry assured him. "I'm in complete possession of all my faculties, and I'm about to accomplish something that has never been done before. In the long and bloody history of our glorious Empire, many have assassinated _one_ Emperor, but I shall be the first to have dispatched a second." He shook his head. "A pity the history books will never give me the credit I deserve."
Gloria was able to look toward Charles. "What is he talking about?"
Charles gritted his teeth and said nothing.
"He never told you?" Larry asked. "Not surprising, I suppose. Still, I think you should know the truth before you die. Do you want to tell her, Charles, or shall I?"
"You bastard!" Charles growled.
"Calling me names will get you nothing," Larry said. He looked at Gloria. "Haven't you ever wondered about the Fifth of October?" he asked her. "Haven't you ever asked yourself why Charles was fortuitously away on Luna when the conspirators were doing their bloody work in Rio? Didn't it ever seem a trifle too convenient? No? Well, no matter. It never occurred to anyone else, either, thankfully."
"I don't believe it," Gloria said flatly.
"You should...shouldn't she, Charles?"
"You incredible son of a bitch! Let us go now and I'll forget about this."
"I think not," Larry replied. "Now, where was I? Oh, yes, the Fifth of October. You see, Gloria, I planned the whole thing. And, of course, Charles was in on it."
"I never did a thing, Glory!" Charles protested. "I swear it!"
"That's true," Larry conceded, "in a sense. I told him what I was going to do, and he never lifted a finger to prevent it from happening. He was content to let events take their course, and even cooperated to the extent of arranging to be elsewhere at the crucial moment. No, he never actually _did_ anything—and that was his crime. My own crime was much more elaborate. Over the course of more than a year, I carefully set the whole thing in motion. I recruited and coerced my assassins, and gave them a vision of the glorious destiny that awaited them once they disposed of the dour and despicable Gregory. The coup would elevate them to power and end the corrupt dynasty of the Hazars, once and for all. I planned it out for them to the finest detail, and convinced them that they would be fools not to play the roles I had assigned them. Ah, it was a true thing of beauty!"
Gloria couldn't believe what she was hearing. Her fear multiplied as she realized that Larry was entirely serious.
"And," Larry continued, "once my cast of conspirators was in place, I assembled a second band of plotters whose task it would be to betray the first. Then—and here is the true genius of the thing—still a third group, to dispatch the second once their work was done. That third group, small and tightly organized, professional killers all, knew only what little I told them and vanished into the depths of the galaxy once their own job was concluded. The result of it all, of course, was a dead Emperor, with two dead sons and three dead nephews, and a stage full of dead and eternally silent conspirators, plus some living but embarrassed Imperial Security men who imagined that it was their own efforts that had aborted the coup. All of which left the throne to none other than the callow youth who now lies beside you, awaiting his own demise. Clever, don't you think?"
Gloria wanted to believe it was a lie, but couldn't. She could just glimpse Charles out of the corner of her eye, his face furrowed in impotent rage.
"Now," said Larry, "you might well ask yourself why I did all of this for the benefit of a young cousin whose company I enjoyed but whose rise to the throne would do me little good. That's a question you might have asked, as well, Charles. If you had any sense, you would have had me killed, but I was reasonably certain that you wouldn't. You're really a little too softhearted for your job. But you were convenient.
"I, of course, could never be Emperor, given my scarlet past. The Council of Lords would never have approved of it, nor would Parliament have confirmed my accession. Gareth was too young, then, so the throne would have passed to Cousin Andrew. He would have made far too clever and competent an Emperor—for my purposes—so it had to be you, Charles. You would serve as a competent placeholder until the time drew nigh for the second act of my little drama. Which it has."
"I almost did have you killed," Charles breathed. "Spirit help me, I should have."
"But you didn't, and now it's too late. We had some fun, cousin, but now it's time for you to die. Gareth, you see, is now of age. I'll step aside, and the Council of Lords will be so relieved to have me out of the way that they'll anoint my son without a second thought. Parliament will go along with it, and the people of the Empire will happily accept Gareth the First as their new Emperor, then get on with their lives."
"You'll never get away with it," Charles said.
Larry shook his head dismissively. "Of course I will," he said. "I did before, and this time it will be much simpler, much cleaner. No messy bodies lying about, no inconvenient doubts about what happened. Oh, I would have preferred not to have done it in my own home, but the opportunity was just too good to let pass. You see, the deaths of Emperor Charles V and his once and future wife will not be the result of assassination. No, it will be the tragic outcome of their well-known propensity for self-indulgence and sexual excess. Show them, Gareth."
Gareth held up a small, flexible bulb with a long tube extending from it.
"You know of Orgastria-29, of course," said Larry. "Well, this dispenser contains a new drug called Orgastria-48. Perhaps you've heard of it. It's not generally available yet because it can have some unfortunate side effects—death being one of them. When people crunch down on a dose of it, the sudden shock to their system, the sudden overload of sensory input, can be fatal. Some sort of neurological spasm, I gather, resulting in the shutdown of the autonomic nervous system. The brain stops functioning, the heart stops beating, breathing ceases, and death results within a few minutes."
Gloria managed not to cry out in joy and relief. She already had the neutralizer! It was right there in her handbag, just a few inches away from her left hip. But her relief died stillborn when she realized that it might as well have been on Earth. She was completely immobilized, and could not possibly reach the lifesaving antidote.
"Just to be sure, what we have here is a concentrated saline solution with what would probably be three or four times the ordinary dose, for each of you. I'm assured that this will be more than sufficient to terminate the vital processes. You see how it will look, don't you? When your bodies are found and examined, it will seem that our randy young Emperor and his famously lascivious ex-wife overindulged themselves, and tragically fucked each other to death. Lovely, don't you think?"
Charles seemed to strain against his invisible bonds but could make no headway. Finally, he relaxed and whispered, "I'm sorry, Glory."
Sorry didn't seem to cover it. Gloria said nothing and concentrated on trying to move her left hand. She managed to flex her fingers slightly, but true motion was still impossible.
"What else?" Larry absently asked himself. "Oh, yes. This room is soundproofed, so it will do you no good to yell or scream—but feel free. It's also shielded against electromagnetic emanations of any sort, so your wrist-coms will be entirely useless. About two minutes after the drug is administered to you, the null field will automatically relax, so that your spasmodic death throes will look realistic. By then, of course, you won't even know what's happening. And, naturally, Gareth and I will be elsewhere at the moment of your death, with plenty of respectable witnesses to attest to our innocence and purity. That, I believe, covers just about everything. Any final words?"
Charles said nothing. Gloria strained against the null field to move her hand, but couldn't.
"No? Good, I wasn't looking forward to hearing them. Well, then, Gareth? Be a good boy and go kill your uncle and his slut."
"Yes, Dad," Gareth said obediently. "Just a second, I gotta adjust the field first." He fiddled with the controls, and Gloria felt a slight contraction of the field. It seemed that the pressure along her left side had lessened, as if she were now on the very edge of the field's restricting influence. With a supreme effort, she managed to move her left hand an inch.
Gareth walked around them, skirting the field, then knelt next to Charles. "You'll never have an easy night's sleep, Gareth," Charles told him.
"Who the fuck cares?" Gareth asked. He reached out and clamped Charles's nostrils together between his left thumb and forefinger. Gloria watched, helplessly, as Charles pursed his lips together and held his breath as long as he could. But eventually, inevitably, he had to gasp for air, and when he did, Gareth quickly stuck the long tube into his mouth and squeezed the bulb.
Charles didn't react for the first few seconds, then suddenly jerked in a convulsive spasm. Only his heels and shoulders made contact with the floor as his body stiffened, arched, and quivered. He gave a strangled cry that sounded like Gloria's name, but then words gave way to mere animal sounds.
Gareth circled around Charles and approached Gloria. She tried to ignore him, ignore the convulsing man at her side, and place herself in a Qatsima mind-set. She retreated inward, reaching out for the center of herself, where time and distance, matter and energy were all one. She felt Gareth squeezing her nose and readied herself for the precise sequence of actions she had plotted. There would be no more than a few seconds to act. When she knew she was as ready as she could make herself, she opened her mouth as if to gasp for air, even though she did not yet need to breathe. She felt the warm, salty liquid spraying against the back of her throat, and at the same instant consciously closed that passage. Another second went by, and she arched her body in an imitation of what Charles had done. Her left hand shot out and thrust into the open handbag. She continued to jerk and writhe in her fake convulsion as she frantically fumbled in the bag for the feel, the shape, of the neutralizing lozenges.
Meanwhile, Gareth had gotten to his feet and made his way around the null field, back to his father. They stood together watching for another moment, then Larry nodded and they left the room, closing the door behind them.
Gloria had to breathe. Had to. As her fingers closed around what she hoped was the neutralizer, she tried to spit the deadly fluid out, but did a poor job of it. She instinctively gasped for air and felt the liquid trickling down her throat. She focused all her concentration, all her strength, on bringing her left hand up to her mouth. It was almost there when the spasm struck. She stiffened and felt a surge of vast, infinite pleasure, far beyond sex or any other sensation she had known or imagined. It pulled her ever onward into a gaping vortex. Not yet, not yet...
Without even seeing what was in her hand, she managed to force the lozenge between her teeth. It might have been the neutralizer, it might have been more Orgastria-48. Whatever it was, she crunched down on it.
The vortex pulled her in...then, just as suddenly, spat her out. She gasped for air, felt it flowing into her like life itself. Her body sagged down to the soft floor, and for a long moment, she simply lay there and breathed.
Charles! Next to her, he was still convulsing, still making urgent, animal noises. She reached into her handbag and came out with another lozenge, but it was Forty-eight. She dropped the lozenge, tried again, and saw that she had the neutralizer. With all the energy that remained in her, she tried to pull her hand up and reach toward Charles. But she couldn't make it. In the center of the null field, the force was simply too strong.
And then the field was suddenly gone. Charles's legs and arms flew out in jerking, electric motions, hitting Gloria hard. She ignored it and pressed the neutralizing lozenge between his lips. But in his spasms, his teeth were clamped shut. She could not get it into him. In an instant of inspiration, she squeezed his nostrils shut, as Gareth had done, and a few seconds later Charles's mouth opened in an instinctive gasp. Gloria shoved the neutralizer in, and his jaws closed on it, grinding it to dust.
LATE THAT EVENING, CHARLES TOOK GLORIA BY the hand and led her out onto the greensward in front of the main building of the Brockinbrough estate. "There's something I want you to see," he told her.
He moved stiffly, painfully, like an old man. His spasms had strained every muscle in his body, and Gloria was gingerly in her own movements.
After the neutralizer had taken effect, they had lain there on the floor of the null-room for what seemed a long time, gradually collecting themselves, slowly coming back from the brink of death. Then they got to their feet, with Gloria supporting Charles and helping him to walk.
Larry was in the main drawing room, standing by a mantel, next to his son. He was holding forth on some trivial topic for a crowd of sycophants when he saw them; his eyes bulged out and his lips moved wordlessly. Charles looked at a black-clad Imperial Security agent, pointed toward Larry and Gareth, and said, "Arrest them." Then he collapsed.
They had hustled Charles into a bedroom, where Gloria told the Security men and the Imperial Physician what had happened. Then she, too, collapsed, more from relief than from the trauma she had endured. A few hours later, both of them were on their feet again.
Together, they went into a guarded room and confronted their would-be assassins. Neither Larry nor his son had anything to say, and even Charles didn't say much.
"We don't want a public scandal," Charles had told them. "It will be simple exile for both of you. You'll take your yacht directly to your lodge on Vymar Three. That's eight hundred light-years from Earth and safely out of the way. You'll have no visitors, and your communications will be monitored. You'll spend the rest of your lives there." He had stared at them for another moment, then turned and walked away from them.
Now, he and Gloria slowly made their way to the fence at the edge of the cliff. Charles pointed to the south, toward the glow of Central. "Any moment, now," he said.
Gloria watched and saw a point of light rising from the city, climbing toward the blackness of space above. It rose steadily for a few moments, then flared into sudden brilliance. A dim spray of sparks trailed slowly downward to the sea.
"It seemed the easiest way," Charles said to her.
Gloria nodded silently.
"Don't look at me like that," Charles said. "It seems to me that you did something similar on Sylvania, didn't you?"
"I didn't say anything," she replied. They stood together at the fence and looked out on the darkened strait.
Finally, Gloria said, "I won't be your Empress, Charles. Not now, not ever."
Charles slowly nodded his head. "I figured as much," he said. "I just hope you can try to understand. I wanted to be Emperor, and I thought I could have it without paying a price."
"There's always a price," Gloria said. She thought of Norman Mingus. She thought of Ed Smith.
"Yes," said Charles. "There always is, isn't there?"
GLORIA SPENT THE NIGHT ALONE IN A BEDROOM at the Brockinbrough estate, then returned to Central the following morning without saying good-bye to Charles. She went up to her suite at the Imperial Cantabragian and found Petra and Jill waiting for her. From their drawn faces and grim expressions, she wondered if they had somehow heard what had happened. No, that was impossible. Then what...?
"Gloria," Petra said softly, "I have some bad news. It's Elaine...she's dead."
Gloria stared at her. The words didn't make sense, didn't register.
"It happened last night," Jill said. "She was at a club, a null-room. Apparently she crunched down on some Orgastria-48. They administered a neutralizer, but somehow, it just didn't work. They couldn't revive her."
"Spirit!" Gloria breathed. Her head spun, her knees felt weak. For a moment, she thought she was going to be sick.
"We've taken care of...of the details," Petra said. "Her family will be notified, and she'll be sent home."
"Eli Opatnu was with her when it happened," Jill said in a flat tone of voice. "In case you want to ask him about it." Gloria looked at Jill and couldn't read the emotions on her face.
"And that brings up something else," Jill said. "I'm sorry to have to spring this on you at a time like this, but I think it's best if I just tell you now. Gloria, I'm leaving Dexta."
"What? But why? Jill..."
"I think I know why you shut down my investigation," she said. "I can add two and two, and I can spell. Eli's strange interest in the case. The way your Erik Manko problem suddenly disappeared. The way Eli kept me from getting a court order immediately, then failed to back me up when I tried to overturn the restraining order. And your sudden turnaround. It all adds up to one thing, Gloria, and you spell it with a Z."
"Jill—"
"Don't try to explain it," Jill said. "I don't want to hear you justifications and rationalizations. I just don't want to hear it. You can't be a little bit corrupt, Gloria, any more than you can be a little bit pregnant. You made your choice, now I've made mine."
"But, Jill—"
"I told you, I don't want to hear it!"
"Dammit, Jill! You're being foolish. Is it worth giving up your Dexta career because of...a minor impropriety? Jill, this is trivial!"
Jill pursed her lips. "Is that how you view it?" she asked. "Is that how you justify it?"
"For Spirit's sake, Jill, what are we talking about here? It's just a low-level scam, a little harmless graft. A few shipments evade taxes, a few Dexta people get paid off. Yes, I know it's wrong, but it's the way things work. Where's the harm?"
"How can you say that after what happened to Elaine?" Jill demanded.
"What? What are you talking about?"
Jill fixed her gaze on Gloria. "Harmless? Gloria, what do you think is _in_ those shipments?"
Gloria opened her mouth, then closed it without saying anything. She stared at Jill and saw the anger and determination on her features. After a long moment, her face softened slightly. "Gloria, Petra," she said, "I'm sorry, but it's over. Good-bye." Jill turned quickly and stalked out of the room.
Gloria watched the door close behind Jill, then slumped down onto a sofa. She put a hand to her head, as if checking to see if it was still properly attached to the rest of her. She saw Petra standing next to her.
"I'm sorry," Gloria said, "but I'm afraid I have more bad news. You're not going to be Lady Petra."
Petra didn't react for a moment, then slowly nodded and managed a wan smile. "Easy come, easy go," she said. She sat down next to Gloria.
"I can't believe Elaine is gone," Gloria said. "If only I hadn't—"
"You can't blame yourself, Gloria," Petra insisted. "Elaine knew what she was doing."
"Yeah," Gloria said. "We all did."
THEY PACKED THEIR BAGS AND TOOK A TRANSIT up to the New Cambridge Orbital Station. Once there, they discovered that no Flyer or Cruiser was available. The departing Dexta bigwigs had taken them all, and Gloria and Petra were forced to book passage home on a commercial liner. The voyage took a week.
It was a mostly silent week. They shared a small stateroom and couldn't avoid each other, but spent most of their time reading or sleeping, or simply staring silently at the walls. Finally, on the fifth day, they began to talk, hesitantly at first, but then more openly, and finally, they told each other everything.
Petra went first, sharing her pain over what had happened with Pug and his family, and then her guilt and self-doubt over her attraction to and involvement with Whit Bartholemew. She felt cheap and weak and stupid, and yet, even now, she still felt a sense of loss and regret about Whit. She wondered what it said about her that she could fall for such a man.
"I did, too," Gloria said. And then she told Petra about Charles, and the rest of it. She recounted her approach to Ed Smith and explained her debt to the zamitat. Once she had started talking, she found that she couldn't stop, and went on to tell the story of Norman Mingus and his family. And Savoy. And finally, the story of Charles and how he became Emperor. For good measure, she added what Charles had done to the Brockinbroughs and what she had done a year earlier on Sylvania.
"Maybe you want out now, too," Gloria said when she had finished her story. "Maybe Jill had the right idea."
Petra shook her head. "No," she said, "I think I'll stay. OSI has already lost Pug and Jill and Elaine. If OSI is going to be the Fifth Quadrant, you'll need me."
"More than ever," Gloria affirmed.
"Gloria? You realize, don't you, that if you hadn't made your deal with this Ed Smith guy, we would never have known that the Savoy shipment was broken down into three loads? And without knowing that, I would never have been able to figure out where the weapons went. And if I hadn't figured that out, maybe I wouldn't have realized the truth about Whit. So your deal with the zamies probably saved a hundred million lives."
Gloria gave her friend a crooked smile. "You're just determined to see the bright side of everything, aren't you?"
"Oh, you know me—Little Petra Sunshine."
"Well, then, allow me to point out that you are much better off without Pug. He was a nice boy, but a boy is what he'll always be. His family will never let him grow up, or grow out of what they want him to be. And as for Whit—Spirit, Petra, the man spent twenty years fooling everyone! Even his mother. In the end, you were smarter than everyone else, and if anyone saved those hundred million lives, it was you."
Petra shook her head emphatically. "No," she said. "He was going to press that button, Gloria. He really was. It was his mother who stopped him, not me."
Gloria thought about that, then said, "I should tell Norman. Maybe Saffron's hundred million somehow balances out Norman's on the family ledger. Maybe they can both find some peace now."
"Maybe we all can," Petra said.
THEY RETURNED TO EARTH AND TRANSITED down to Dexta Headquarters in Manhattan. But instead of heading for the street to walk back to their building, Gloria paused at the turnoff to the Transits to Brooklyn. "I'm going to spend a few days at my place on Long Island," she explained.
"Good idea," Petra said.
"Say hi to your mom for me. Tell her for me that she ought to be very proud of her daughter."
Petra grinned at Gloria. "Even if I'm not Lady Petra of Weehawken?"
"Hey, kiddo, you're a Dexta Twelve now! What more could anyone want from life?"
GLORIA QUICKLY MADE THE HOP TO BROOKLYN, picked up her Ferrari skimmer in the Dexta lot, and made her way home. Along the way, Billie Holiday kept her company, singing "God Bless the Child." She pulled into the carport of her house in the dunes, wandered aimlessly through the empty house, then went out to the pool and found, to her surprise, that she had a new tree growing in her yard. An exceptionally ugly tree. A note was pinned to it.
She opened the note. "The Imperial Gardener informed me that a simple cutting wouldn't work, so we brought in a whole tree from Belonna Five," it read, in Charles's flowing handwriting. "I hope it does you some good."
Gloria smiled and looked up at the gnarled, bony limbs, the ungainly proportions, and the bare branches. There were no leaves yet, but the roots looked strong and tenacious.
She walked over the dunes and down to the beach. The tide was low, and she had to walk a long way to reach the water's edge. She stared out to the gray, indeterminate horizon for a long, silent time, thinking no particular thoughts. Then she reached into her pocket and pulled out a handful of purple lozenges. She stared at them for a moment, then threw them as far as she could, into the cold ocean waters. Then she turned and walked back across the dunes and sat down to think and remember, in the comforting shadow of her glashpadoza tree.
Gloria Van Deen may have achieved her Fifth Quadrant, but can she stand up against an alien civilization which is determined _not_ to become part of humanity's empire?
Don't miss Gloria's next exciting adventure in
**BURDENS
OF EMPIRE**
by
**C.J. RYAN**
Coming in Fall 2007 from Bantam Spectra
Here's a special preview:
**BURDENS
OF EMPIRE**
On Sale Fall 2007
LORD KENARBIN CUT A SPLENDID FIGURE AS he stepped out onto the dock, and he knew it. He was tall and trim, strikingly handsome, with medium-length silver hair curling over the tops of his ears and piercing blue eyes that commanded the attention of all who fell under their gaze. His strong, slightly bony nose suggested Hazar blood, while his smooth, swarthy complexion implied a complex genetic heritage. In his ninety-seventh year, he looked as virile and vigorous as a man half his chronological age.
Kenarbin carried himself with a diplomat's aplomb and a drill-sergeant's precision. His shiny black knee boots, form-fitting white breeches, and gold-trimmed, deep-blue tunic were accented by the diagonal red sash draped across his torso, signifying his Imperial mandate. His features automatically assumed a familiar, well-practiced mien of amiable determination and boundless self-confidence. He paused and stared into the middle distance for a few moments in order to let the swarm of media imagers record his arrival.
Aside from the gaggle of media reps and the cluster of official greeters, both human and native, there was not a lot to see. His Cruiser had splashed down in a broad, sluggish river, brown and oily—the local Mississippi or Amazon, he supposed. The dun-colored landscape offered little in the way of vegetation or relief, and the chill, steady wind sweeping in from the river felt unfriendly and forbidding. The sky was cloudless but yellowed from its cargo of dust and debris, and the single cold star provided a weak, unflattering orange radiance.
In the distance, the dark towers and crenellated walls of the city looked medieval, and the smaller structures dappling the plain could have been the huts and hovels of serfs. A patina of age clung to the place, a reminder of the weary millennia of experience boasted by this civilization, which had achieved star-travel when humans were still scrimmaging with Neanderthals and mammoths. Yet it was this world that had been conquered and occupied by the upstart humans and their burgeoning Empire—an outcome emphasized by the sheltering canopy of military vehicles that patrolled the ugly sky above.
_Denastri,_ he thought. Well, he'd seen worse.
Kenarbin took it all in, then turned to face his welcoming committee and offered them a hearty smile. It was met by unsteady grins from the humans and blank, impassive gazes from the indigs—Empire slang for indigenous species. The Denastri, he had been told, were not a demonstrative race, and the expressions on their alien faces might have meant anything at all, or nothing.
_We are not welcome here._
The unavoidable thought did not trouble Kenarbin unduly. Humans weren't really welcome in most places they went. It didn't matter. The Empire was here, and it was here to stay. It was Kenarbin's job to get the locals to accept that immutable fact. _They don't have to like us, and we don't have to like them._
Lord Kenarbin had been coming to places like this for more than half a century, representing the Empire with skill and imagination. In the process, he had become something of a legend, having pulled Imperial fat from fires that might have consumed lesser negotiators. His reputation was well and justly earned, and if the job had become familiar from repetition, it remained a point of pride to do it to the best of his considerable ability. These days, Emperors used him sparingly, recognizing that his very presence magnified the significance of any mission on which he embarked: Kenarbin was here because Denastri was important, and Denastri was important because Kenarbin was here.
Three years earlier, in a swift and relatively bloodless little war, the Imperial Navy had smashed the small, antique Denastri fleet, putting an abrupt end to thirty thousand years of conflict within the minor grouping of stars known to Terrans as the McGowan Cluster. While the local tides swept endlessly back and forth between the Denastri and their neighbors, a millennium of relentless human expansion had finally brought the Terran Empire to the doorstep of the McGowan Cluster, 1,053 light-years from Earth, and henceforth the locals would have to behave themselves. The backwater world of Denastri, and everything on it or under it—particularly the latter—now belonged to the Empire. His Imperial Highness Charles V had decreed peace, and peace there would be.
Some of the locals had refused to believe or accept this turn of events, and even the presence of a division of Imperial Marines had failed to convince the holdouts. If anything, the sputtering insurgency had picked up steam in the preceding year, making life uncomfortable and dangerous for the Terrans who had come here for the sake of Imperial power and corporate profits.
The indigs, in any case, were a fractious lot, split three ways and as eager to slaughter one another as their human overlords. Instead of meekly bowing before the overwhelming might of an Empire that spanned two thousand light-years and encompassed 2,673 worlds with a population exceeding three trillion, some of them remained determined to fight on, heedless of the consequences for themselves or their lackluster little world. Kenarbin had come to reason with them.
Sanjit Blagodarski, the Imperial Governor, stepped forward and extended his right hand. Kenarbin clasped it in both of his.
"Welcome to Denastri, Lord Kenarbin," said the Governor.
"Thank you, Governor. Good to see you again, Sandy. You're looking well."
His first lie, less than a minute after setting foot on the planet. In fact, Blagodarski looked awful. Drawn and frazzled, he seemed to have aged twenty years during the ten since they had last met. The Governor shrugged off the obvious falsehood with a weak smile and introduced his Imperial Secretary, a Level XII Dexta functionary named Freya Benitez, and the commanding officer of the Occupation Task Force, General Steven Ohashi. The general gave Kenarbin a crisp military nod along with a firm handshake. "Glad you're here, Milord," said Ohashi. That, in itself, struck Kenarbin as an ominous note. Marines were seldom happy to see diplomats on their turf.
"And now," said Blagodarski, "it is my privilege to present the Premier of Denastri. Honored Premier, may I present Lord Kenarbin?"
The alien stepped forward and extended a four-fingered hand, which Kenarbin took in his. Its flesh felt cold.
"Vilcome to our furled," said the Premier, with obvious effort.
"Thank you, Honored Premier," Kenarbin replied as he stared into the dark vertical slits of the alien's eyes. The creature was vaguely humanoid—two arms, two legs, nearly as tall as Kenarbin. But its face was narrow and noseless, with large, drooping triangular ears, sallow skin, a sharp, pointed chin, and a mouth that would have looked at home on a rainbow trout. The vertical almond-shaped eyes seemed to be all pupil, and looked like the entrances to shadowy, unexplored caverns. From a narrow, bony crest at the top of its head sprouted a long shank of blue-black hair, braided and bound with thin colored threads. The Premier's clothing consisted of a belted saffron-colored robe that fell nearly to the ground.
Kenarbin released the Premier's hand and touched a stud on his tunic, activating the translation software on the computer pad in his pocket. "Honored Premier," he said, "I bring sincere and heartfelt greetings from His Imperial Highness, Emperor Charles V." He paused to let the Premier's own pad translate his words into the fluid, tonal language of the Denastri, then continued. "The Emperor has asked me to convey his deep personal gratitude for your service to the Empire, and to your world. He expresses his confidence that, working together, we shall restore peace and prosperity to his loyal subjects on the rich and beautiful world of Denastri."
He paused again as the Premier absorbed the translation. Kenarbin studied the Premier's face carefully, but could detect no identifiable reaction. After a moment, nictitating membranes closed in from the sides of the Premier's eyes in an approximation of a blink. Then the Premier spoke in a flowing, almost musical passage that was pleasant but incomprehensible to human ears.
The computer rendered the translation in a soft, precise, androgynous voice. "You are mostly kind to be here, generous Lord," it said. "The words of Imperial Highness Fifthborn Charles are registered in deep appreciation by this humble Thirdborn. Peace and prosperity inspire all to high wishfulness. It is a goodness."
Kenarbin frowned and furrowed his brow. He had been warned that the translation software was still a work-in-progress, but he had hoped for something better than this.
"A goodness, indeed, Honored Premier," Kenarbin said. "I look forward to working with you to make it so."
"Yes," the Premier responded. "Work will make good. We will build again that which has fallen and return— _unknown word—_ to Denastri and the felicitations of Fifthborn Charles and his grasping Empire. Yes."
Kenarbin glanced at the Governor, who tilted his head a little and offered a wan smile. "You'll get used to it, Milord," he said.
"I think I know what he's saying," Kenarbin said. "I just wish I could be sure that he knows what _I'm_ saying."
Blagodarski shrugged. "We manage," he said. "For the most part. We should be on our way now, Milord. We'll have you safely into the Compound in a few minutes. It's not wise to linger too long in an exposed position like this."
"It isn't? Why not?"
"Because we make too good a target, Milord," General Ohashi explained.
"It's not as bad as it sounds," Blagodarski hastily added.
"The hell it isn't," Ohashi mumbled under his breath. The Governor gave him a sharp, reproving glance, but Ohashi looked away, focusing his gaze on the far side of the river, as if searching for snipers.
Kenarbin nodded. "I see," he said. "In that case, gentlemen, ladies, Honored Premier, perhaps we should continue our discussion in the Compound. I look forward to seeing your capital city, Honored Premier. I understand that it is older than any on Earth."
"Earth is young," the Premier agreed. "Denastri is blessed with the continuing wisdom of all our time. Perhaps you will learn— _unknown word—_ from us, Lord Kenarbin. A goodness."
"Undoubtedly," said Kenarbin. "A goodness."
The party began moving along the dock. Kenarbin noticed that the dock was flanked by squads of armed, helmeted Marines, who snapped to attention as he passed. Ahead, surrounding a small fleet of limo skimmers, the Marines were accompanied by what appeared to be native Denastri troops, hefting Terran plasma rifles. They were noticeably taller than the Premier and their skin was more orange than yellow.
"Fourth- and Fifthborns," Blagodarski said as they walked. "Warrior caste."
Kenarbin nodded. "Fine-looking troops," he said to the Premier, who seemed momentarily confused by the comment and didn't respond immediately.
After a few moments, the Premier said something that the computer rendered as "Beauty is in the eye of the beholder."
Kenarbin, surprised, looked at Blagodarski. "The software has trouble with clichés," the Governor explained. "Garbage in, garbage out, I suppose. Not that what you said was garbage, Milord."
"Perish the thought," Kenarbin replied with a chuckle.
At that instant, a dazzling burst of intense white light blinded him. A split second later, he was deafened by the thunderous crack of a concussion device. Stunned and all but senseless, Kenarbin felt hands grabbing at him, clutching his arms, and dragging him, then lifting him. He flailed out uselessly and shouted something equally useless, then felt himself being thrown bodily into what must have been the backseat of one of the limo skimmers. Someone shoved him down onto the floor of the vehicle, and he felt it lifting and moving.
_Security,_ he thought. _They're getting me out of here._
Sound and sight gradually returned. He tried to turn over and push himself up, but found himself being pushed back down against the floor. "What's happening?" he demanded, his own words sounding faint and distant. He could hear no response, and saw nothing but a blurred smudge of maroon carpeting, an inch from his nose.
Kenarbin calmed himself. There had been other attempts on other worlds, and he knew the routine. Security people would treat him like a sack of highly valuable potatoes until they were certain that the threat had passed. Annoying but necessary. He could hear the high-pitched whine of the skimmer now, competing with the ringing in his ears. He wondered if the Governor and the Premier were safe.
Minutes went by, and he felt the lurching, darting progress of the skimmer. It seemed to him that they ought to have reached the Compound by then. He managed to twist around a little and turned his head to look up. He expected to see burly Marines on the seat above him. Instead, he found himself looking into the narrow, orange-tinted face of one of the Denastri warriors.
"What's happening?" he asked again. "Where are you taking me?" The Denastri offered no response. Possibly, he had no translation device and didn't understand.
Kenarbin again tried to push himself up from the floor, but the warrior rudely shoved him back down. The first tickle of fear and suspicion began to dance at the edges of his mind.
"Dammit, what the hell is going on here?"
The Denastri leaned forward a little and stared down at him. The alien eyes looked placid and unsympathetic. "You is ours," it said in Empire English.
"What? What do you mean by that?"
"Vord is 'hostage,' yes?" the warrior asked.
Comprehension flooded into Lord Kenarbin in a cold, unwelcome wave.
"Yes," he said at last, "that's the word."
THE SUN GLARED IN NORMAN MINGUS'S FACE, bright enough to be annoying, even through the polarized panoramic dome. Poised just above the irregular peaks on the north rim of Shackleton Crater, its unrelenting radiance was an imposition on an old man's eyes, and gave Mingus yet another reason to resent the necessity of these semi-annual excursions to the South Pole of Luna. He envied Charles, sitting opposite him on the far side of the three-tiered circular amphitheater where the Imperial Oversight Committee was pleased to hold court. The solar inferno was comfortably positioned behind the young Emperor's right shoulder, and he had to face only the less constant, if much closer, fires of angry Parliamentarians.
The Empire was in trouble. Mingus had known it for decades, but had hoped—naively, perhaps—that the inevitable reckoning could be postponed beyond his time. Let the next generation deal with it. His own generation had seen enough sorrow and tumult and, Mingus believed, deserved a respite. He had labored to see that they got it, and his efforts had resulted in a half-century of relative peace and prosperity.
But the Terran Empire, like others before it, was a victim of its own success. With the conquest of the Ch'gnth Confederacy in 3174, the last remaining external threat to the Empire had been removed. For a thousand light-years beyond Imperial space, in every direction, no existing power was capable of thwarting the continued expansion of the Terrans. There were, to be sure, a few minor impediments, like the fledgling Gumnaki Hegemony, waiting a couple of hundred light-years beyond the Frontier in Sector 4. The Gumnaki—a race of vast pretensions and minimal subtlety—had carved out a mini-empire of forty or so worlds, but they posed no immediate threat. The Empire would have to fight them someday, Mingus presumed, but probably not on his watch. The outcome, in any case, would be a foregone conclusion. The same applied to the handful of other races that might object to the onward march of _Homo sapiens_.
Paradoxically, it was the very absence of an external threat that now imperiled the Empire. Many historians, Mingus knew, held that it had been the rise of Islam in the seventh century that had forced the consolidation and forged the power of modern Europe. It was not until after the temporary decline of Islam that the Europeans, deprived of a strong external enemy, fell upon themselves in two centuries of fratricidal insanity, polluting the historical record with names like Napoleon and Hitler, Verdun and Auschwitz. The later fall of the Soviet Union had much the same effect on the Americans, who built their doomed empire without ever admitting to themselves what they were doing. The Terran Empire, at least, harbored no illusions about what it was and what it meant to achieve.
But without the balance and focus provided by external powers, the whirling centrifugal forces inherent in so vast an empire were bound to tear it apart someday. Mingus, a lifelong student of history, knew that the larger an empire grew, the harder it was to govern. He also understood mathematics. The old Earthly empires only had to deal with two dimensions, but the Terran Empire was condemned to grapple with three. Thus, the Empire, 2,000 light-years in diameter, comprised some 4.2 _billion_ cubic light-years of space, with a surface area on the Frontier of 12 million square light-years—all of which had to be patrolled and policed. If the task was not inherently impossible, it was certainly daunting.
Mingus accepted the implications of the unforgiving math but Charles, alas, did not. Charles was the third Emperor he'd had to deal with during his time in office, and by far the most difficult. Bumbling old Darius had paid little attention to the niggling little details of his realm, which was probably just as well. Gregory hadn't been around long enough to make any difference. But Charles, now in his seventh year as Emperor, was young, arrogant, and ambitious. At thirty, he was finally showing some signs of maturity; perhaps the recent birth of Henry, his son and heir, had something to do with that. Yet his essential character was unlikely to change, and Mingus knew that at his core, Charles was a cold, ruthless son of a bitch. _Takes one to know one,_ Mingus wryly conceded. Age might mellow Charles, but it was not likely to improve him.
As the meeting droned onward with the dramatic speed and force of an oncoming glacier, Mingus focused his attention on Charles, who wore an expression of intense and genuine pain on his handsome face. Emperors, like other men, were capable of the most basic of human emotions—even love.
Two years earlier, Charles had attempted and failed to win the return of his ex-wife, the glamorous and popular Dexta official Gloria VanDeen. Mingus didn't know the details of her rejection of his offer of remarriage and elevation to Empress, but he had sensed a profound change in Charles following the episode. He could have any woman in the Empire except for the one he wanted most, and the realization must have seared his soul in some deeply painful way. The private man, Mingus supposed, must have surrendered some essential part of himself to the public figure; Charles had been thwarted, but Charles V could not be. Under pressure to produce an heir, he had wanted Gloria for his Empress and the mother of his son. Denied his desire, he had swiftly turned in another direction and taken Lady Patricia Kenarbin as his Consort. Not Empress, but Consort. Some Emperors never married at all, and were content to sire their successors with a Consort; Charles, it seemed, would be one of them. He didn't need an Empress; he didn't need Gloria; he didn't need love.
And yet, he seemed to have found it anyway. Mingus could not gauge the depth of Charles's feelings for Lady Patricia, but he suspected that it was considerable, and had probably come as a surprise even to Charles. It happened that way sometimes, for some men. And whatever his feelings for Patricia, there was no doubt about his intense attachment to young Henry.
The last time Mingus had seen them together, the Imperial Heir had cutely puked on the shoulder of his Imperial Dad. Charles had merely laughed, and Mingus, a veteran of five marriages and many children, considered it the most human moment he'd ever seen Charles experience. For a moment, Mingus almost liked him.
Now Lady Patricia was in anguish over the fate of her father, and Charles the man was clearly at odds with Charles the Emperor. What Charles would gladly do for the woman he loved, Charles V could not do. Man and Emperor were trapped together in a golden web that circumscribed his actions as surely as it bound his heart. Charles was no less a hostage than Kenarbin, and no conceivable negotiation could free him.
AS THE MEETING FINALLY BROKE UP AND PEOPLE drifted off to the buffet, one of the Emperor's aides approached Mingus; Charles wanted to have a word with him. Mingus found him standing alone on the upper tier, staring down into the perpetually shadowed depths of Shackleton Crater. Ice-mining machinery was moving around on the floor of the crater like aimless glowworms.
"How is Lady Patricia holding up?" Mingus asked him.
Charles glanced up. "She tries to put on a brave face, but I know it's killing her. She's not used to this sort of thing. She always led a pretty sheltered life at Court."
"She's Bill Kenarbin's daughter," Mingus pointed out. "She'll be as strong as she has to be."
Charles nodded. "She's only twenty-one," he said. "Sometimes I wonder if I did her a favor by choosing her."
"I'm certain that she thinks you did, Charles. What's more, you did yourself a favor."
The Emperor smiled a little. "Even a blind hog finds the occasional acorn," he said. The smile faded. "Norman? I need to ask you for something. Something difficult."
"I'm at your service, Highness."
"Maybe not this time. I want him back, Norman. I want my son to have a grandfather, and Trish to have a father. This is tearing me up inside. I've never..." He trailed off and shook his head.
"It's different when you have a family," Mingus said.
"Isn't it just? I had no idea, I truly didn't. I never really knew my own parents. Just as well, probably. My father was an idiot, and my mother...well, you knew her."
"It isn't easy, being a Hazar. They meant well, Charles."
"I don't want somebody telling Henry the same thing about me someday, Norman. He was a bastard, but he meant well. Spirit! It's not enough just to _mean_ well. I want to _do_ well. For both of them."
"What do you want from me, Charles?"
The Emperor looked the Dexta Secretary in the eye and said, "I want you to send Gloria to Denastri and get him back."
Mingus was genuinely shocked by the Emperor's words. He wanted to ask if he was serious, but there was no doubt that he was.
"I know what you're going to say, Norman. You can't do it officially. I understand that. But, dammit, Gloria gets _results_!"
"She does," Mingus agreed.
"I wouldn't ask this if there were any other way. You know that."
"I do." Mingus returned Charles's steady gaze, and for a few moments, neither of them spoke.
Charles ran a hand through his dirty-blond hair and shook his head. "She's infuriating and impossible, but somehow she always gets the job done. She wouldn't do it if I asked her, but maybe you can find some appropriate excuse to send her to Denastri. From what I hear, the Dexta staff there are at one another's throats. Isn't that the kind of thing she's supposed to handle? Strategic interventions, and all that?"
"It is," Mingus said, giving a small nod. "To tell you the truth, I've even considered it myself. She might be able to knock some heads together and at least get everyone on the same page. But as for Kenarbin...I can't ask her to do that, Charles. Not officially, and not even unofficially."
"Well, could you at least _suggest_ it?"
"When a Dexta Secretary makes a suggestion, a Level IX can hardly avoid taking it as an order. No, Charles, I can't ask her, and I won't suggest it. There's only one person who can ask her."
Charles nodded and took a deep breath. "Then," he said, "I suppose I must."
**About the Author**
C.J. Ryan lives and works in Philadelphia.
ALSO BY C.J. RYAN
**DEXTA**
**GLORIOUS TREASON**
THE FIFTH QUADRANT
A Bantam Spectra Book / October 2006
Published by
Bantam Dell
A Division of Random House, Inc.
New York, New York
All rights reserved
Copyright © 2006 by C.J. Ryan
* * *
Bantam Books, the rooster colophon, Spectra, and the portrayal of a boxed "s" are trademarks of Random House, Inc.
* * *
www.bantamdell.com
eISBN: 978-0-553-90296-9
v3.0
| {
"redpajama_set_name": "RedPajamaBook"
} | 6,883 |
Q: Tesla AC Motor Patent: Is this waveform 90 degrees out of phase in the diagram? I'm referring the following pictured patent:
Specifically, this portion of the patent:
When flipping and superimposing the two magnets in the picture below, in the middle of the image I see two wave forms 90 degrees out of phase, with acceleration. I'm aware an accelerating magnetic field induces the current due to the Lorentz force:
I'm curious if this means anything, is meaningless, or was Tesla's way of showing the physical laws in the patent?
A: It's meaningless.
The illustrators that are used for drafting the final published form of a patent usually redraw anything the inventor has provided. This is why more or less all published patents from any given country have the same sort of look and feel to their drawings.
The 'waveforms' you are looking at are an illustrators' convention that there is some bulk material beyond what has been shown, which has been 'torn off' to leave ragged edges. While that on the south pole could look regular enough to be intended as a waveform, the north pole edge is much more irregular. To me, the overlapping 'waveforms' look to be nearer to antiphase (180 degrees) than 90.
I think if Tesla had been intending to leave an Easter Egg in the patent, they would have been better waveforms, and in quadrature phase.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,626 |
{"url":"http:\/\/motls.blogspot.com\/2012\/10\/eu-nobel-peace-prize.html?m=1","text":"## Friday, October 12, 2012\n\n### EU: a Nobel Peace Prize?\n\nI am left speechless by the information that the EU has won the Nobel peace prize (Reuters; Nobel prize website). After Arafat, Gore+IPCC, and Obama, among a few others, it's yet another incredible choice. The 2012 Nobel prize in literature was just fine \u2013 a Chinese writer may be official but he's still successful in the world and the success isn't due to his OK links with the Chinese communist leadership. But a Nobel prize for the EU?\n\nPicture from Market Watch, \"EU\u2019s Peace Prize and the mockery of timing\" (click at it)\n\nIt's not even clear what it means for the EU to win the prize. What is the EU? I mean: Who is the EU? I am surely the EU. I've voted to incorporate my country into it and I became one of the 450 million citizens of this confederation. Is the prize supposed to be picked by some officials like the ex-commie and future-fascist Barroso? Why is he more representative of the EU than myself, for example? Or at least my favorite members of the European Parliament (who are surely against most things that Barroso stands for)? What the hell does it mean in the context of the EU that claims to be a democracy where people are created equal?\n\nOne thing is that I don't understand what it means for a confederation of countries \u2013 that, according to some of the naive proponents, even wants to become a single federal country \u2013 to win a Nobel Peace Prize. Why hasn't the U.S. won the prize? Or Czechoslovakia, for the Velvet Divorce? Or other countries?\n\nAnother question is why exactly the EU. In the recent two years, the EU has been the most important source of tension in the world among comparably large regions. The main source of the tension \u2013 the fiscal problems that spread everywhere \u2013 are exactly the attempts to unify countries that can't be united because they're very different, and exactly this source of global tension seems to be what the Nobel Peace Prize was awarded for. The EU is getting a previously prestigious prize for having destroyed the stability of the global economy, for destroying trillions of dollars in people's investments across the world, for suppressing the legitimate rights and freedoms of nations and individuals on the old continent, and for bringing the nations to hate each other, and all these \"achievements\" are being legitimized by the Nobel committee in the name of peace.\n\nAs I wrote many times, it's not even true that the European integration \"caused\" peace after the war. The causal relationship was the other way around: nations began to trade and be friends with each other because they no longer found fighting constructive. Also, the peace persisted for such a long time mainly because of fears of nuclear destruction that a conflict between European powers could lead to. So the U.S. and the USSR should really get the Nobel prize for the Cold War if this is the justification.\n\nJust unbelievable \u2013 nevertheless, the crazy decision was actually unanimous. The Nobel Peace Prize has become a giant joke. Every decent citizen of a European nation should spit on every member of the Nobel Peace Prize committee whom she or he meets on the street.\n\nOne hour before the announcement, a Norwegian TV channel leaked the information about the 2012 Peace Nobel Prize winner. Czech president Klaus was reported to say that he didn't believe the information and he considered the report of the Norwegian TV to have been a joke. Internet polls at leading Czech news servers show that about 80% of Czechs disagree with the choice of the winner.\n\nAfter the announcement, Klaus answered the journalists here:\n\nLet me translate it for you:\nJournalists: How would you react to the Peace Nobel Prize for the EU?\n\nKlaus: Well, I made my first reaction around 10:56 or 10:57 am when it was just an internet report and I honestly thought it was a canard, a hoax, or a prank because even in my wildest dreams I couldn't imagine someone who meant it seriously. And this was my response before the official announcement. When I talk after the event, let me state that I think it is a tragic blunder.\n\nThe Nobel prize has always been meaningful because it was given to a human, to an individual, to a well-defined person for a particular achievement \u2013 a unique achievement. To reward an institution, in this case a bureaucratic institution, is an empty honor. I am not sure whether someone will even have the courage to dare to accept this prize. So please, this is probably not a recipe how to give the Nobel prizes.\nWithin 12 hours, the discussion at iDNES under this video attracted about 2,000 comments.\n\nWe have regional elections in Czechia and I am going to vote now.\n\nAlso, Iraq decided to buy 28 pieces of L-159 A, mostly new copies (4 old, 24 new) of a Czech combat aircraft, for 1 billion dollars. I wonder whether the folks who claim that an \"imperfect\" contract that a defense minister does and that removes a few million dollars from the treasury means \"corruption\" note that in this case, they actually added this impressive contract. A corruption may only occur if the corrupt person does or may actually benefit in some way. Anything else is just the inefficiency of the government \u2013 one of its basic and most universal properties.\n\nA Capuchin joins the 99% movement\n\nPeter F. sent this hilarious video showing a South African monkey who joined the Occupy Wall Street movement, unlike its friend:\n\n1. I happen to reside in the EU country of Norway the most narcissistic country in the world.\nYou don\u2019t know that Norway is an EU country?\nWell you right, apart from the fact that EU laws written by the EU bureaucrats have precedence over Norwegian laws trough the so call EES treaty.\nI wouldn\u2019t be to surprised if at the next year these clowns are going to give the Nobel Peace Price to the Kingdom of Norway or maybe they give it to Gro Harlem Bruntland, the Agenda 21 and UN global warming wacko.\n\n2. Hi, just a technicality. Norway may be a member of many European things but it is surely not a member of the European Union.\n\nhttp:\/\/en.wikipedia.org\/wiki\/European_Union\n\n3. Yes technically you are right. However the weird thing is that EU laws through the EES treaty is law in Norway, while whatever one can say about the democracy of the EU, the electorate in Norway has nada-say how these laws are made.\nHow this works is that while Norway is outside the EU and protect their expensive agriculture, fish industry and make sure the EU has no access to the oil revenue everything else is under EU roles.\nQuite a weird construction.\n\n4. Unbelievable! Are you sure that the Reuters story isn't a plant by the Onion? :)\nMaybe next year's Peace Prize will go to Gaia.\n\n5. Czechia and Slovakia are, together, deserving of a Nobel peace prize. I can think of no better examples than the Velvet Revolution and the Velvet Divorce for settling deep differences peacefully.\n\n6. Gordon, I propose to give it to the universal Female gender next time. After all women have rarely directly initiated any wars...\n\n7. I don't understand how they can justify that the EU did something for peace? Membership in the EU does not guarantee at all that if e.g. France decides to attack England that all other EU countries will support England. NATO does that: the reason behind peace in Europe is mostly the fact that European countries are members of NATO and not the EU. Furthermore, the current way of EU's approach to handling the huge debts of some of its countries, like Greece, have shown that Europe has not learned anything from its own history. Forcing Greece to stay in the Eurozone by demanding harsh austerity measures increases hate of Greek people towards other countries (especially Germany). In this respect the situation is very similar to the one of Germany after the WWI which led to the escalation of extremist parties and subsequently to WWII. The same things are happening in Greece right now! No, there is absolutely no contribution of the current EU to peace...\n\n8. Who will receive the prize money? Some central EU bank? Nobel Peace Prize is losing its significance by these repeated awards to individuals\/entities that are not deserving. It's become a wishful thinking award.\n\n9. The next logical step is that they will award the Nobel Peace Prize to the Nobel Peace Prize jury themselves for awarding the Nobel Peace Prizes over the years... ;)\n\n10. ...posthumously ;-)\n\n11. I'm still shocked and possibly will so say, and so are several people I talked to.\n\n12. Czech president V\u00e1clav Klaus calls it a mistake, error and fault (also a joke) - to translate the word \"omyl\" correctly.\nI take it he will never run for EU office ... :-)\n\n13. Next year they had better give the Nobel Prize committee IQ tests or else we will end up with someone like Cornel West being \"honored\". I am sure they left the \"Ig\" prefix off of the announcement.\n\n14. I saw this story this morning, and assumed it was a joke. The reality is quite bizarre. I guess this action, in conjunction with the committee's giving B.O. the Peace Price for having done absolutely nothing, means that the Nobel Peace Price Committee is the WORLD'S BIGGEST JOKE.\n\n15. Next year I'm nominating \"Peace\" for the Nobel Peace Prize.\n\n16. The next step is obviously to give the 2013 award to the Taliban--for their commitment to destroying priceless historical, cultural and artistic relics and their furthering of humanitarian murder.\n\n17. Brian G ValentineOct 13, 2012, 4:22:00 AM\n\nI wonder how Alfred Nobel would feel today, knowing he left his entire fortune to a collection of crackpots and Communists\n\n18. The Nobel Peace Prize has become such an absurd farce that it is utterly disgusting. Next time around they will award it to the Cat in the Hat.\n\n19. The only rational explanation I can think of is that Al Qaeda terrorists have sleepers amongst the Norwegian Prize Committee. To root them out, we have to get each committee member to draw a cartoon of the Prophet :)\nReally though, this is sad. The prize has been so demeaned that it should be dropped entirely.\nEven Obama was totally mystified when he was awarded the prize, and after Gore got one, I did not think it could get any more insane.\n\n20. It's not just an omyl. It's a tragic blunder: :-)\n\nOn the other hand, I guess that he would accept a high position of the EU if he could be given one, and reformed Europe back to a Europe of sovereign nation states.\n\n21. Sadly, this year's IgNobel prizes did not get featured on The Reference Frame, but the Nobel Peace Prize took care of that by its voluntary self-destruction.\n\nFor 2013, I am nominating this paper for Nutrition: a study of last meals ordered by prisoners on death row, which shows that last meals are shockingly unhealthy :)\n\n22. Apologies for the omission, Eugene. I now consider the Ig Nobel Prize and the Peace Nobel Prize to be equally serious - and the Peace Nobel Prize is linked to a greater amount of money which is why I wrote about this one. ;-)\n\n23. Nice, JIm, still a discrimination against physics. What about a prize for equilibrium? Wherever there is equilibrium, wars stop and everyone lives in peace.\n\n24. Sort of an admission by the committee that there have been nobody worth pointing out, promoting peace and harmony the last several years.\n\nHarmony, harmony (hmm...). Another word for equilibrium.\nI nominate for next years award, interference patterns.\n\n25. Could have been worse Lubos. They might award EU Nobel Prize in Economic...\n\n26. Ha, what about Helen of Troy? But your proposal is a good one Shannon, I support it for 2014. Next year peace Nobel prize should go to my cat, a very peaceful and contemplative creature who has never harmed anyone or anything (excepting mices, rats, moles, birds and the occasional rabbit or bat). I will make sure that the prize money is put to good use..\n\n27. They have awarded themselves into irrelevancy. I'm okay with the NPP becoming and international joke. Maybe while the world is laughing at them it will be, for a short period of time, a little better.\n\n28. Hahaha! The Nobel committee must have inhaled too much CO@!\n\n29. I propose an award for the wave function. They could simultaneously award it and not award it. Perhaps the idiots on the committee thought they were doing just that - after all the EU is also about to collapse.\n\n30. Are you going to go and accept your prize in person?\n\n31. A Russian video on the peaceful \"living together' in France... thanks to the EU.","date":"2013-05-22 19:03:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.21840421855449677, \"perplexity\": 2746.332120510661}, \"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-2013-20\/segments\/1368702298845\/warc\/CC-MAIN-20130516110458-00020-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}\label{sec:intro}
\SW{
To maintain an environment map in dynamic environments, a robotic visual SLAM system must be capable of detecting changed objects on the map. To be effective, a map maintenance capability must fulfil two requirements: the robot must know where it is (i.e., map-relative robot self-localization), and it must differentiate between changed objects and nuisances (i.e., robot-centric change detection). Motivated by the recent maturity of self-localization techniques (e.g., visual place recognition, loop-closure detection, map-matching), we propose the reuse of self-localization capability to address the latter requirement (i.e., change detection).
}{
}
\SW{
Recent studies demonstrated that self-localization performance is a very useful measure of the likelihood of change (LoC) for change detection. The experience-based mapping framework originally proposed in \cite{churchill2013experience}
is a good example. In that framework, an environment map was permanently augmented by multiple sub-maps (i.e., visual experiences) using data from differing environment conditions. If self-localization with such a map performs sufficiently well, it covers the encountered conditions and there is no need to update the map (i.e., low LoC). However, if self-localization performance is poor, the map does not cover the encountered conditions during the sortie (i.e., high LoC).
}{
}
\SW{
Such a new change detection scheme, referred to as ``detection-by-localization" in the current study, has two main advantages:
\begin{enumerate}
\item
No additional storage or detector engine (but only existing map database and localization engine) is required;
\item
The degration of map quality (i.e., need of map update) in terms of self-localization performance can be directly detected.
\end{enumerate}
}{
}
\SW{
This paper addresses a new detection-by-localization task that detects changes at subimage- or object- levels (Fig. \ref{fig:a}). Unlike previous methods that classify a given query image as ``change" or ``no-change", we not only classify a query image but also detect changed objects (in the form of bounding boxes) within the image frame.
The relationship between image- and object-level change detection is analogous to the relationship between image \cite{sivic2003video}
and object localization \cite{jiang2012randomized} in the computer vision literature.
Object-level detection becomes an even more challenging task than image-level detection, owning to the requirements of object segmentation and low image resolution.
Importantly, object-level change detection is common in robotics
and has many important applications,
including
patrol robot in partially changing environments \cite{drews2010novelty}
and
object segmentation via change detection \cite{finman2013toward}.
However, object-level change detection has not been explored in the context of detection-by-localization.
}{
}
\SW{
In our task scenario, a ranked list of reference or background images retrieved from a map database by a self-localization system is viewed as the measure of self-localization performance. Formally, we consider a detection-by-localization scenario in which reliable viewpoint measurement (referred to as ``ground-truth" viewpoint) is available \cite{churchill2013experience}, and we measure self-localization performance (i.e., LoC) by the rank value assigned to the ``ground-truth" reference image. The self-localization model is inspired by our previous work on unsupervised part-based scene modeling, in which the popular bag-of-visual-features (BoVF) self-localization scheme \cite{sivic2003video} is extended in two ways.
\begin{enumerate}
\item
Instead of point features (e.g., SIFT), discriminative subimage-level features (referred to as ``scene parts") are used to model a query/reference scene.
\item
Reference subimages are retrieved using query subimages, and are aggregated into an image-level decision (i.e., ranked list).
\end{enumerate}
In our contribution,
\begin{itemize}
\item
we explain how such an image-level self-localization model can be adopted to the novel task of subimage-level change detection,
\item
by addressing how a ranked list can be interpreted as a subimage-level pixel-wise LoC map,
\item
and by exploring ways of aggregating subimage-level LoC maps into the final decision of a single image-level LoC map.
\end{itemize}
Our unsupervised ranking based approach is an advantage, because it does not require training examples on the image collection.
This is very important, because we must deal with increasingly complex maps and images, and automatic collection of their ground-truth data by the robot-self is not a trivial task.
}{
}
\editage{
\figA
}{}
\SW{
Presently, most image change detection algorithms \cite{ccr1} focus on pairwise image comparison, to differentiate a given query-background-image-pair. It is straightforward to adopt such a method to object-level change detection by introducing an image segmentation preprocessing step \cite{im2008object}. This has recently achieved state-of-the-art performance with a weakly-supervised setting \cite{aaai17cd}. However, these image-to-image differencing methods are not directly applicable to real-time SLAM systems. First, they are too expensive (on average) to compute in real-time. Moreover, they require memorization and maintenance of many background images proportional to the map size. Some alternative popular approaches are those that classify a sole query image instead of a query-background-image-pair as a change or a no-change, compared to an offline pretrained background model. Such an approach has the potential to realize model compactness by using compressed background models such as BoVFs, geometric model and compact manifold learning. However, its storage overhead is proportional to map size and increases in unbounded fashion. Moreover, updating this background model significantly complicates the map updating process. To suppress such computational time and space complexities by reusing existing resources of map database and localization engine is an objective of the current study.
}{
}
\SW{
}{
}
\SW{
}{
}
\section{Detection-by-Localization Framework}
\SW{
We consider a new detection-by-localization framework
consisting of a self-localization system and a change detection system, in which self-localization performance
(i.e., rank value of the ``ground-truth" reference image as explained in Section \ref{sec:intro})
is used as an LoC measure for change detection. Both self-localization and change detection can be confused by differing fields-of-view (FoV), occlusions, and seasonal changes of object appearance or appearing of visible objects.
}{
}
\subsection{Scene Model}\label{sec:sm}
\SW{
The scene model is assumed to be a BoVF scene model \cite{sivic2003video}. The vast majority of real-time visual SLAM systems (i.e., iBOW-LCD \cite{Garcia-Fidalgo2018}) employs BoVF models.
Whereas other scene modeling schemes such as
the deep ConvNet classifier have been studied in the self-localization literature.
The BoVF scheme is advantageous to computational speed, compactness, and discriminativity, and has been a de facto standard method for real-time self-localization tasks, such as loop-closure detection.
}{
}
\subsection{Self-Localization Model}
\SW{
The self-localization system is modeled as a ranking function.
That is,
it aims to output a ranked list of all reference subimages in descending order of similarity from a given query subimage.
If necessary, each query subimage is resized to an appropriate size before being input to the self-localization system.
This model is valid for most self-localization systems,
which are based on image retrieval formulations.
}{
}
\SW{
Not only are rank lists output by self-localization algorithms, but some relevance scores for individual reference images (e.g., TF-IDF, conditional probabilities, maximal consistencies) are also output. However, it is not straightforward to interpret such relevance scores to LoC values within a detection-by-localization framework. We reserve this issue for future work.
}{
}
\SW{
Image-level and subimage-level rank values are available as output of any self-localization system.
The BoVF-based self-localization systems first evaluate subimage-level similarities between each query subimage and each reference subimage,
and then aggregate their similarity values to obtain the image-level similarity scores or rank lists.
It is straightforward to output rank lists
in the order of subimage-level similarity
for individual query subimages.
}{
}
\subsection{Change Detection Model}
\SW{
The change detection model follows the standard formulation of image change detection \cite{ccr1}. Given a query image, it aims to estimate a pixel-wise LoC map of the image, which is then converted to a pixel-wise binary change mask.
Its performance is evaluated in terms of 101-point (i.e., 0.00, 0.01, $\cdots$, 1.00) interpolated average precision (AP) \cite{yolo}.
}{
}
\editage{
\figB
}{}
\section{Object-Level Change Detection}
\SW{
This section provides detailed methods
for object-level change detection
via detection-by-localization (Fig. \ref{fig:b}).
}{
}
\subsection{Scene Modeling}
\SW{
The scene modeling segments a given query/reference image to yield a pool of useful scene parts (in the form of bounding boxes) or subimages, so that the segmented subimages remain consistent between the reference and an unseen query images. This problem is referred to as ``consistent part segmentation" \cite{kanji2015unsupervised}. This problem is significantly ill-posed, owning to differing FoVs, occlusions, and seasonal changes of object appearance. Even state-of-the-art segmentation algorithms are far from perfect, producing several false positive parts. We address this issue by hypothesizing a relatively large number of subimages and verifying them in the spirit of majority voting \cite{jiang2012randomized}, as we explain in Section \ref{sec:dbl}.
}{
}
\SW{
}{
}
\SW{
We use both unsupervised and supervised segmentation techniques,
inspired by our previous work \cite{iv18sugimoto}.
Unsupervised segmentation techniques (e.g., BING \cite{cheng2014bing}) provide category-independent object proposals, even for objects with unseen classes.
We use a set of five pre-defined bounding boxes for every image.
For an image with width $w$ and height $h$, these five bounding boxes are defined as
$[w/3, 2w/3]$$\times$$[h/3, 2h/3]$,
$[0, 2w/3]$$\times$$[0, 2h/3]$,
$[w/3, w]$$\times$$[0, 2h/3]$,
$[0, 2w/3]$$\times$$[h/3, h]$, and $[w/3, w]$$\times$$[h/3, h]$.
Supervised segmentation techniques (e.g., YOLO \cite{yolo}) provide more precise object regions supported by rich semantic information for objects with known classes. We use YOLO with threshold value of 0.05. It should be noted that whereas supervised techniques are trained on pre-defined object classes, it is often useful to propose unseen objects,
having a visually similar appearance to pre-defined objects.
Both supervised and unsupervised proposals are beneficial,
because objects with both pre-defined and unseen classes can be changed objects.
}{
$[w/3, 2w/3]$$\times$$[h/3, 2h/3]$,
$[0, 2w/3]$$\times$$[0, 2h/3]$,
$[w/3, w]$$\times$$[0, 2h/3]$,
$[0, 2w/3]$$\times$$[h/3, h]$,
$[w/3, w]$$\times$$[h/3, h]$
}
\subsection{Self-Localization}
\SW{
In \cite{Merrill2018RSS}, a novel AE-based ConvNet for loop-closure
detection was presented.
It was designed to be an unsupervised, convolutional AE
network architecture, tailored for loop-closure, and
amenable for efficient, robust place recognition.
Its performance was evaluated via
extensive comparison studies of the
deep loop-closure model against state-of-the-art
methods on different datasets.
Additionally, the histogram of oriented gradients (HoG) was employed to compress images while preserving salient features and projective transformations (i.e., homography).
In contrast,
we are interested in
the basic effectiveness of an AE-based self-localization,
and we
implement
the convolutional AE in a rather simplified setting without using
a HoG -based extension.
Our AE consists of three convolutional layers and each convolutional kernel size is 3$\times$3,
employing max-pooling, batch-normalization and ReLU activation for each layer.
Each input subimage is resized to 256$\times$256,
and mapped by the AE to a 16,388-dim feature vector.
}{
max-pooling, batch-normalization
256$\times$256
16388
}
\subsection{Detection-by-Localization}\label{sec:dbl}
\SW{
Following our previous research in \cite{kanji2015unsupervised}, we modeled individual scene parts belonging to different modalities, and adopted rank fusion techniques, derived from the field of multi modal information retrieval (MMR) \cite{atrey2010multimodal}. MMR techniques are originally designed to deal with increasingly complex document collections, corresponding to subimage or part collections in our application domain and queries, consisting of not only text modalities but also non-textual modalities such as visual words in image retrieval, geo-tags, user rate, etc. More formally, they aim to fuse multiple retrieval results and queries from different modalities into a single ranked list to make a final decision. Unsupervised approaches requiring no training data are desirable, as explained in Section \ref{sec:intro}. Such unsupervised MMR approaches are broadly classified into two categories: early fusion \cite{yan2006probabilistic} and late fusion \cite{cormack2009reciprocal}.
Early fusion aims to fuse multiple queries from multiple modalities at the level of the input feature descriptor. This approach is advantageous in exploring correlation between multiple data modalities. However, it requires multiple queries to be converted to the same format prior to fusion. This is not applicable to a wide range of self-localization applications.
Late fusion aims to fuse multiple queries at the output decision level (e.g., the level of relevance score or rank list). This approach is further divided into score fusion \cite{montague2002condorcet} and rank fusion \cite{cormack2009reciprocal}.
Score fusions rely on the so-called raw-score-merging hypothesis and fuses relevance scores from multiple queries. However, as a key limitation, it requires each modality's retrieval to output a relevance score, significantly limiting its application.
However, rank fusion aims to fuse rank values from multiple queries. As a key advantage, this approach requires only ranked lists (commonly output by self-localization), and importantly it can outperform the score fusion approach in previous applications \cite{hsu2005comparing}.
}{
}
\SW{
Based on these considerations, we chose the rank fusion approach as our basis. In our previous study, the rank fusion approach was explored in a different context of image-level self-localization \cite{kanji2015unsupervised}. The current study
is based on this previous method, but with key differences:
\begin{enumerate}
\item
The previous study aimed at image-level ranking, whereas the current study aims to obtain subimage-level pixel-wise rank values.
\item
The previous method took as input non-overlapping query subimages (from color-based segmentation), whereas the current study takes relatively large numbers of overlapping query subimages (from unsupervised/supervised object proposals).
\end{enumerate}
To address this issue, we must deal with a novel task of pixel-wise rank fusion.
Considering the new setting where each pixel may belong to multiple subimages (referred to as qBBs) as in Fig. \ref{fig:b}, we fuse them all to obtain the rank list for each pixel. Because the number of such qBB is different among different pixels, it is not straightforward to compare the fused rank list between them. Our previous ranking method in \cite{kanji2015unsupervised} assumed that the number of rank lists was pre-defined and fixed. Thus, it was not directly applicable to the current problem. We now address this issue by introducing a new parameter, $N(i)$, the number of times a document appear in the rank lists, as suggested in \cite{imhof2018study}. The modified rule for rank fusion takes the form:
\begin{equation}
R(i)=N(i) \times \sum_{k=1}^{N(i)} \frac{1}{R_k(i)},
\end{equation}
where $N(i)$ is the number of the $i$-th document appearing in the rank lists, and $R_k(i)$ is a rank value (of the ground-truth reference image) assigned by the $k$-th query's retrieval. If $N(i)$ is a constant, the function $R(i)$ reduces to the previous fusion rule. Our algorithm takes as input a collection of query bounding boxes (qBBs), and performs the following steps:
\begin{enumerate}
\item
Each of qBBs is checked, and if they overlap, the intersection areas are computed and registered as a new qBB; This process is repeated until no new qBBs are found.
\item
For each $i$-th qBB, the number of overlaps, $N(i)$, is computed and related ranked lists are assigned, and then, the assigned ranked lists are fused by the function $R(i)$.
\end{enumerate}
}{
\begin{equation}
R(i)=N(i) \times \sum_{k=1}^{N(i)} \frac{1}{R_k(i)},
\end{equation}
}
\section{Experiments}\label{sec:exp}
\SW{
To test the performance of the proposed method, we performed extensive experiments on change detection, using the publicly available NCLT (North Campus Long-Term) dataset \cite{nclt}. For each test data, we compared our pixel-wise rank fusion method with score fusion methods, which rely on the availability of relevance scores. Note that this score fusion method can be viewed as an adaptation of previous non-ranking-based (anomaly-based) change detection methods \cite{christiansen2016deepanomaly} to our problem domain. We used it to evaluate the level of achievement of the proposed method.
}{
}
\SW{
The NCLT dataset is a large-scale, long-term autonomy dataset for robotics research collected at the University of Michigan's North Campus by a Segway vehicle robotic platform. The data we used in the research includes view image sequences along vehicle's trajectories acquired by the front facing
camera of the Ladybug3 with GPS. From the viewpoint of change detection benchmark, the NCLT dataset has desirable properties:
\begin{enumerate}
\item
It involves both indoor and outdoor change events during seamless indoor and outdoor navigations of the Segway robot.
\item
It contains not only typical changed objects such as cars and pedestrians, but also various kinds of changes such as building construction, construction machines, posters, tables and whiteboards with wheels.
\item
It has been recently widely used in robotics communities as experimental benchmarks for various tasks, such as self-localization \cite{jmangelson-2018a}.
\end{enumerate}
In the current study, we use four datasets
``2012/1/22", ``2012/3/31", ``2012/8/4", and ``2012/11/17"
(referred to as WI, SP, SU, AU)
collected across different four seasons.
Fig. \ref{fig:env} shows
the experimental environment
and examples of changed objects
in the dataset.
}{
}
\editage{
\figC
}{}
\editage{
\tabA{}{}
}{}
\SW{
We formulated change detection as a binary classification problem. Each test sample consists of the pairing of two images from two different seasons, referred to as query and reference (or background) images. To create a test set, each image is viewed as a query image candidate and is paired with a reference image with a nearest neighbor viewpoint,
forcing it to belong to a different season
whose viewing angle must be nearer than
45 degree from that of the query image. The test set consists of positive and negative sets. A positive set consists of positive image pair samples, whose query image contains changes with respect to the counterpart reference image.
A negative set consists of those image pairs with no change.
For the positive set, these changed objects are manually annotated in the form of bounding boxes.
The total number of positive and negative samples are 1,054 and 4,188. Image size is 1,232$\times$1,616. Note that successive frames in the NCLT dataset often contain identical and visually similar changed objects in successive frames. To make our test set contain various changed objects, we sampled at most one query image from such successive frames that contain such a visually very similar object. Thus, we obtained 1,054 positive image pairs from the 4$\times$3 season pairs. Fig. \ref{fig:env} shows the robot's viewpoint trajectories and representative changed objects in the test set.
}{
}
\SW{
For the test system, the visual appearance of each subimage is described by the AE as a feature vector. Feature vectors are indexed and retrieved by a nearest neighbor engine. Prior to the indexing, every feature vector is L2 normalized. For the nearest neighbor, the L2 distance is used as dissimilarity metric. Two independent engines are then employed for the two different object proposal methods (i.e., supervised and unsupervised methods). Most time consuming part of the test system are YOLO-based object proposal and AE-based part feature extraction, which were 11.1 msec per image and 0.9 msec per subimage (Geforce GTX Titan). The number of subimages per image was 15.9 $\pm$ 3.0 (mean$\pm$std), and thus total time cost was 11.1 + 0.9 $\times$ 15.9 = 25.41 msec.
}{
}
\SW{
Note that the length of the rank list can differ among query subimages. Therefore, raw rank values are not comparable among different queries. To address this issue, we normalize the raw rank values by the rank list length to the [0,1]-interval. In our preliminary experiments, we also tested two variants of the proposed system:
\begin{enumerate}
\item
One where the rank was not normalized, and
\item
Another where reference subimages were sorted not in descending order but in ascending order of the LoC prior to the rank fusion.
\end{enumerate}
We found that both variants did not work well and yielded significant performance drops (e.g., AP $<$ 0.6). Moreover, we tested another variant where SIFT's Harris-Laplace region and descriptor were used in place of qBBs and the AE descriptor, finding such a variant also does not work well. The reason might be that such a small Harris-Laplace region could not provide contextual and semantic information that are rich enough for change detection.
}{
}
\SW{
Difficulty indices play an important role in object detection literature standardizing benchmark datasets \cite{radenovic2018revisiting}. In the current study, we introduce two types of difficulty indexes: {\it RoC} and {\it SoB}. {\it RoC} (rate of change) is defined as the area of changed object (i.e., overlap region between the qBB and the ground-truth change qBB) normalized by the area of qBB. If {\it RoC} value is too small for a given qBB, it is difficult to detect such a small changed object. Thus, difficulty of change detection will increase.
Effectiveness of this overlap-based difficulty index was verified in our previous study on self-localization \cite{kanji2016self}.
{\it SoB} (size of bounding box) is defined as the area of qBB normalized by the area of image. If the {\it SoB} value is too small for a given qBB, such a subimage looks very dissimilar from the original non-cropped full image,
and the probability of obtaining inappropriate self-localization result will be higher, because typical self-localization systems implicitly or explicitly assume that image size is the same or similar between the input query image and the corresponding reference image. We sampled a test query image set so that query images in the positive and negative sets respectively have restricted {\it RoC} values within their respective $[RoC_{min}^{ + }, RoC_{max}^{ + }]$
and
$[RoC_{min}^{ - }, RoC_{max}^{ - }]$,
and restricted {\it SoB} values within
their respective
$[SoB_{min}^{ + }, SoB_{max}^{ + }]$
and
$[SoB_{min}^{ - }, SoB_{max}^{ - }]$.
By default,
we set
$[RoC_{min}^{ + }, RoC_{max}^{ + }]=[0.9, 1]$,
$[RoC_{min}^{ - }, RoC_{max}^{ - }]=[0, 0.05]$,
$[SoB_{min}^{ + }, SoB_{max}^{ + }]=[0, 0.4]$,
and
$[SoB_{min}^{ - }, SoB_{max}^{ - }]=[0.4, 1]$.
}{
$[RoC_{min}^{ - }, RoC_{max}^{ - }]$,
$[SoB_{min}^{ + }, SoB_{max}^{ + }]$
$[SoB_{min}^{ - }, SoB_{max}^{ - }]$
$[RoC_{min}^{ + }, RoC_{max}^{ + }]=[0.4, 1.0]$
$[RoC_{min}^{ - }, RoC_{max}^{ - }]=[0, 0.4]$,
$[SoB_{min}^{ + }, SoB_{max}^{ + }]=[0.9, 1.0]$
$[SoB_{min}^{ - }, SoB_{max}^{ - }]=[0.0, 0.05]$.
}
\SW{
We create various test sets with different levels of difficulty by using {\it RoC} as a difficulty index. More formally, the difficulty of a given test query image set is controlled by two thresholds $RoC_{min}^{ + }$ and $RoC_{max}^{ - }$. If, the {\it RoC} value for a qBB exceeds the threshold of $RoC_{min}^{ + }$ then the qBB is assigned the ground-truth change label of ``change".
If, the {\it RoC} value falls below the threshold of $RoC_{max}^{ - }$ then the qBB is assigned the label of ``no-change". Other qBBs assigned neither ``change" nor ``no-change" are not considered as members of the test set. We created a collection of test sets with different levels of difficulty by changing these two threshold values.
}{
}
\SW{
We compared six different change detection schemes: ``rank fusion", ``rank w/o fusion", ``rank fusion ($\le 2$)", ``rank fusion ($\le 3$)", ``score max", and ``score sum". ``Rank fusion" is the proposed algorithm. ``Rank w/o fusion" is a variant that differs from the proposed algorithm because it does not fuse qBBs to create new qBBs. Therefore, it does not perform pixel-wise rank fusion. ``Rank fusion ($\le k$), $k$= 2, 3" are motivated by the question ``how many pixel-wise rank fusions are required to obtain optimal performance?". Unlike the proposed algorithm, they perform pixel-wise rank fusion at most $k$-times per pixel. If there exists more than $k$ overlapping subimages for a pixel, $k$ subimages are randomly selected and used for that pixel. ``Score max" and ``score sum" are adaptations of ``score max" and ``score sum" strategies in MMR \cite{atrey2010multimodal} per our application domain: the pixel-wise LoC map. These differ from the proposed algorithm, in that relevance scores are used in place of rank values (i.e., assuming the availability of relevance scores output by the self-localization system). For the relevance score, we use L2 distance of feature vectors. The ``score max" fuses pixel-wise relevance score values $v_1$, $\cdots$, $v_k$ by fusion rule
\begin{equation}
v=\max_{i=1}^k v_i.
\end{equation}
The ``score sum" fuses them by fusion rule
\begin{equation}
v=\sum_{i=1}^k v_i.
\end{equation}
}{
}
\SW{
We investigated
change detection performance for different $RoC_{max}^{ - }$ values. Generally, if a changed object looks small, its change detection will be more difficult. This experiment is motivated by the question ``how small an object can achieve acceptable change detection performance?".
Table \ref{tab:a}
shows the AP performance for different $RoC_{max}^{ - }$ values. All the methods tested are not sensitive to the parameter values. Overall, the ``score max" strategy yielded the best performance, whereas it requires the availability of relevance scores output by the self-localization system and relies on the raw-score-merging hypothesis. The proposed ``rank fusion" strategy yielded comparable performance, despite it not requiring relevance scores. It clearly outperformed the ``rank w/o fusion" strategy and other variants.
}{
}
\SW{
}{
}
\SW{
}{
}
\SW{
}{
}
\SW{
In summary, the proposed method successfully detected visually small objects from small (i.e., low-resolution) subimages. Moreover, change detection performance was higher when pixel-wise rank fusion was used. The proposed rank-fusion scheme is comparable to score-fusion schemes, which require the availability of relevance scores. Whereas the current study focuses on combining AE based feature vectors with pixel-wise rank fusion, in a future work, we plan to explore other self-localization schemes, such as different features (e.g., hand-crafted features), and heterogeneous self-localization systems (e.g., particle filters).
}{
}
\section{Conclusions and Future Works}
\SW{
We presented a new detection-by-localization method for object-level change detection using a ranking-based self-localization model, a novel model to evaluate the likelihood-of-change (LoC) of a given query subimage, and unsupervised rank fusion. The ranking-based model did not require training data, nor did it rely on raw-score-merging hypotheses. We consider the proposed unsupervised framework as beneficial to many visual SLAM systems, because such an unsupervised model can generalize it to a broad class of self-localization systems. Future work will investigate ways to use computational resources of different self-localization systems for large scale change detection and map maintenance applications. We will analyze its asymptotical behavior when the number of map databases and self-localization engines increases.
}{
}
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,933 |
Ušće (; pronounced ) is an urban neighborhood of Belgrade, the capital of Serbia. It is located in Belgrade's municipality of Novi Beograd. Ušće is located on the mouth of the Sava river into the Danube, thus the name (ušće is Serbian for confluence). It occupies Novi Beograd's Blocks 10, 13, 14, 15 and 16 on the Sava's left and the Danube's right bank, covering a tip of land that overlooks the islands of Little War Island and Great War Island to the north and the old core of Belgrade, the fortress of Kalemegdan to the west. Ušće borders the neighborhoods of Staro Sajmište and Savograd on the south. As a compact grassy and forested area it stretches along the bank of the Danube into the Block 10, to the Zemun municipality and the Hotel Jugoslavija and the ENJUB shopping mall.
Spanning over , Park Ušće is the largest official park area in Belgrade.
History
Nica Beach
A sandy beach with the cabins, kafanas and barracks, used as sheds by the fishermen, occupied the area of the modern quay, north of the Branko's Bridge. It was one of the favorite vacation spots of Belgraders during Interbellum. People were transported from the city by small boats and the starting point was a small kafana "Malo pristanište" in Savamala. Occupying the left bank of the Sava, it was on the location of the access ramp for the future King Alexander Bridge. The objects were demolished manually, including numerous kafanas: "Ostend", "Zdravlje", "Abadžija", "Jadran", "Krf", "Dubrovnik", "Adrija", etc. The only one that wasn't demolished was "Nica", predecessor of the modern Ušće restaurant. In total, 20 proper objects and 2,000 cabins, barracks, sheds, etc. were demolished, jointly by the municipalities of Zemun and Bežanija, which owned half of the land each, and the proprietors of the objects. The plan was to build an embankment instead.
However, the beach itself survived the construction of the bridge in 1934 as it only made access easier. By this time, "Nica" became the largest city beach. It was filled with fine, white sands and was shallow. In time it stretched even further along the Sava, up to the location of the modern Gazela Bridge. There was a sandy barrier island (sprud) close to the central section of the Sava in front of the beach so it was a popular goal to be reached by the swimmers. The beach was finally closed in 1938 when the construction of the embankment began. The barrier island was destroyed after 1948, when the sand was used to fill the marsh during the construction of New Belgrade. The beach itself was called Nica (Serbian for Nice, in France) after one of the kafanas.
Characteristics
Like all of Novi Beograd, Ušće is flat, and without buildings to hide that fact like in the rest of the municipality, that is quite obvious here. With only three buildings and several smaller edifices, Ušće is the least urbanized section of Novi Beograd but some residential blocks are administratively attached to the local community of the same name, which had the population of 9,991 in 2011.
Ušće is a vast grassy and forested area along the river banks. As such, it is used by many Belgraders as a recreational area or as a place for organizing political gatherings or musical concerts. However, many areas are not cultivated, but left to grow wild. During the high levels of the Danube and the Sava, bank areas are always flooded.
As maintained open parkland with high levels of human activity, Ušće is characterized by low diversity of breeding landbirds, but in winter the banks are very popular among birdwatchers as it is possible to observe rare and interesting waterbirds swimming at the confluence off the banks of Great and Little War Island. Ušće is one of the most important eBird hotspots in Belgrade with its own webpage: Ebird Hotspot
In September 2018, Belgrade's mayor Zoran Radojičić announced that the construction of a dam on the Danube, in the Zemun-New Belgrade area, will start soon. The dam should protect the city during the high water levels. Such project was never mentioned before, nor it was clear how and where it will be constructed, or if it's feasible at all. Radojičić clarified after a while that he was referring to the temporary, mobile flood wall. The wall will be high and long, stretching from the Branko's Bridge across the Sava and the neighborhood of Ušće, to the Radecki restaurant on the Danube's bank in the Zemun's Gardoš neighborhood. In the case of emergency, the panels will be placed on the existing construction. The construction is scheduled to start in 2019 and to finish in 2020.
Park Ušće
Much of Ušće already is, or is projected to be, a large park. On 24 August 2017, the mayor of Belgrade, Siniša Mali, announced a massive works in the park which would, as he described it, look like a Central Park in New York City or a Gorky Park in Moscow. The entire project, which would arrange the area of between the Branko's Bridge and Hotel Jugoslavija, is based on the ideas of Jan Gehl. The entire project is to be finished by 2020. Park will be divided in 7 zones. Two zones closer to Zemun and Zemunski Kej will be "Movement" and "Nature" .
They will be mostly nature and sports oriented with various sports terrains and children playgrounds. Next zone "Serbia" ("Serbia 1" and "Serbia 2" ), will be adapted for the promotion of Serbian culture, folklore, gastronomy, etc. The next two zones are "Music" and "Water", while the two closest to the Branko's Bridge, surrounding the Museum of Contemporary Arts, are "Art" and "Science". They will cover and some works in it started immediately. For 2018, Mali announced the construction of the gondola lift, which will connect the geographical tip of Ušće with Kalemegdan, and of a giant Ferris wheel across the Great War Island. He confirmed the erection of a tall flagmast.
The flagmast was already brought up before. Without any previous public mention of the project, Mali announced on 26 June 2017 construction of the flagmast in Ušće, which would hold the Serbian flag, by . He already declared it a symbol of Belgrade which would bring many tourists who will take photos with the mast, which is "important for the entire Serbia" and will make Belgrade a modern and more beautiful city. He also added that the project is an "initiative of president Aleksandar Vučić".
Mayor Siniša Mali claimed the money for the project, 200 million dinars, has already been marked in the city budget, due to the previous "savings". Asked how is that a priority when one third of Belgrade still has no sewage system, Mali said that both the sewage and the mast are needed. Dragoljub Bakić, member of the Serbian Academy of Architecture, said that the city government treats Belgrade as the "prey" they won, labeling their politics as an "urban reality show as they bomb us every day with some other nonsense". Bakić also criticized the parceling and concreting of the park area.
Rodoljub Šabić, state Commissioner for the information of public importance, publicly criticized the project on Twitter and was slammed by the author of the idea, president Vučić on the press conference who said that Šabić is "against the Serbian flag" and that "those who never do anything, criticize all the time". Šabić replied that Vučić always subjectively and without any criteria defends people from his close circle and that being against the pole which would cost 1,65 million euros is not the same thing as being against the Serbian flag. He added that it would be better to buy something meaningful, like the MRI for some hospital.
Mali announced the expansion of the project. Apart from the one tall, whose price in few days grew to 230 million dinars (1.9 million euros), 30 additional flagmasts, each high, are planned. Three of them will be placed on ten entry points into the city. They will cost additional 134 million dinars (1.1 million euros), so 31 flagmasts will cost the city almost 3 million euros. Reminded that for that money a kindegraten or a school could have been built, Mali said "don't your worry about it" and that we are "rich enough country and rich enough city", repeating Vučić's claim that those who are against the project actually don't like the Serbian flag. Three days later, Mali changed his statement and said that the idea is not Vučić's but is actually a joint decision of the city and the state.
Construction of the gondola lift was confirmed in March 2018 when the idea of a pedestrian bridge was dropped after it has been described as "complicated" and "unstable". On the Kalemegdan side, the station will be dug into the hill, below the fortress' Sava Promenade. On the Ušćе side, the starting point will be next to the Skate Park, across the Ušće Tower. The entire route is long, of which will be above the Sava river itself. Estimated cost is €10 million and duration of works 18 months, but it is still not known when the construction will begin. The criticism of the project continued, from the officially used name (gondola instead of a traditional Serbian žičara) and chosen location, to the route, especially the Kalemegdan station. Several reports were filed against the people involved in the project, but the state didn't act upon any of them. The project is opposed to the laws and city's decisions on protection as it will have up to high pillars which would obstruct the protected view from the Belgrade bridges towards the fortress and from the fortress to New Belgrade. Also, the Kalemegdan station is to be dug into the walls of the fortress, which is protected by the law as the Cultural Heritage of Exceptional Importance.
Even larger works were announced in December 2018. The area for the reconstruction was enlarged to , including parts of both existing Ušće and Friendship parks. The height of the envisioned Ferris wheel was set to and construction of a new building, the 21st Century Arts Museum was planned. The new museum will have of exhibition space, four times more than the existing Contemporary Arts Museum, though it shouldn't be taller (). Further additions to the project are the Natural Science Museum (floor area , height ) and an aquarium with the area of . The international architectural design competition for the entire project was scheduled for 2019.
Cutting of over 100 trees in the park, because of the gondola lift began in March 2019. With an enlarged price of €15 million and unified opposition to the project by the environmentalists, architects and urbanists, with additional cutting of 40 trees across the river, on the Fortress, this prompted popular protests. Citizens organized and as city was cutting the trees, they were planting the new seedlings instead.
City administration declared the reconstruction finished in March 2019, though nothing of the planned attractions was built. It was also announced that the Ferris wheel project was scrapped while the construction of the flagmast was pushed to the end of the year. In May 2019 the idea of the Ferris wheel was revived, with the possible addition of an alpine coaster and a roller coaster. In February 2021 it was announced that city planned the construction of the flagmast in 2021, with the price lifted to 269,2 million dinars (€2.3 million euros). Though nothing has been done regarding the flagmast, a certain amount of money was spent from the city budget every year on it, including the projected 2022 budget.
A monument to Rabindranath Tagore was dedicated on 13 July 2021, close to the Branko's Bridge. The verses of Tagore's poem Where the mind is without fear are carved on the pedestal. The memorial also marked 95 years since Tagore's visit to Belgrade in 1926, when he was a visiting scholar at the University of Belgrade.
Friendship Park
Friendship Park () is a section of the park bounded by the Nikola Tesla Boulevard on the south and the esplanade along the Danube on the north. It covers an area of and is specific for the fact that most of the trees are planted by the politicians, artists and celebrities. It was founded in 1961, during the 1st Summit of the Non-Aligned Movement in Belgrade. The first seedling was planted by the Serbian politician Voja Leković but followed by Josip Broz Tito and Jawaharlal Nehru.
In the next decades they were followed by over 180 people, including Gamal Abdel Nasser, Haile Selassie, Sirimavo Bandaranaike, Archbishop Makarios III, Elizabeth II, Mikhail Gorbachev and Ion Iliescu, who planted the last "celebrity" tree in 1993 and the park deteriorated due to the lack of maintenance. It was renovated in the mid-2000s and in 2007 the initiative was started to restore the tradition of planting the trees. In 2007 each tree got a memorial plaque with the names of the persons who planted it and the new seedlings were planted by the members of the EYOF and the members of the Rolling stones. Projects for the park caused controversies since the early 2000s.
On the one-year anniversary of the end of the NATO bombing of Yugoslavia in 2000, the monument Eternal Flame was built as a pet project of Mirjana Marković, the wife of Slobodan Milošević and later the stir was caused by the proposal for construction the building of the Belgrade Opera in the park. Idea was later dropped. As of 2018, the Eternal Flame monument still has no necessary permits and is technically illegally built.
An area of within the park, the Peace Alley, has been declared an . The park was fully renovated from September 2018 to March 2019.
Features
Buildings
Ušće has only three buildings, but two of them are monumental. One is the Palace of Serbia, formerly Palace of the Federation (Serbian: Палата Федерације, Palata Federacije), a seat of the former federal governments of Yugoslavia and Serbia and Montenegro (also called the SIV building). Another monumental building is the Ušće Tower, with 134 m(antenna) of height almost tallest building in the Balkans (it was second tallest before the NATO attack on Serbia in 1999 when Avala TV Tower just outside Belgrade was razed). Third important building is the Museum of Contemporary Art, close to Sava river's left bank.
In January 2018, it was announced that the construction of a twin building to the Ušće Tower, a Tower II, will start in February. The foundation stone was laid on 27 February 2018. The building is 22-stories high and the entire project cost €65 million. It was officially opened on 11 June 2020.
The modernist sculpture dedicated to the "Women's courage" was placed on the plateau in front of the Ušće Tower on 4 December 2019. The sculpture is work of Marko Lađušić.
Restaurants
Other facilities in the neighborhood are several restaurants along the river banks. Oldest and the best known is the restaurant "Ušće". It was built in 1960 by Stojan Maksimović and entered Serbian textbooks of architecture as the first public facility of the contemporary architecture in Belgrade. The interior design was work of Mario Maskareli. Due to its location, near the confluence and close to the river with the view on Kalemegdan, Cathedral Church and the old section of Belgrade across the Sava river, the restaurant was featured in numerous movies, music videos and broadcasts and until 1990s was one of the most distinguished restaurants in town. The restaurant was refurbished and ceremonially reopened on 1 June 2017 under the name of "Nacionalna Klasa". Only five days later, on 6 June, the restaurant was partially burned in the fire.
Sava's left bank contains numerous barges (Serbian: сплав, splav), which since the early 1990s became center of the famed Belgrade's night life. When the complete reworking of the park was announced in 2018, it was said that all barges between the Branko's Bridge (Sava river) and the Military Complex (arm of the Danube across the Great War Island) will have to be moved. That would basically leave the entire neighborhood without restaurant barges. First barge was placed in 1991. "Lukas", though opened in 1985 on another location, by the mid 1990s became extremely popular and a gathering place for the members of two criminal clans, Voždovac and Zvezdara. Numerous shootouts, which included the river police and fatalities, ensued. Today, the splav with different name is on its former location. One of the first to play turbo-folk music, "Lukas" has been described as the "monument to the 1990s". By 1992, almost a dozen of new barges appeared "overnight", including a very popular "Mozzart", which sank after 2000. This entire section of Ušće is today recognized as a symbol of the "splavovi-culture".
Quay
The embankment along the rivers has been turned into the long promenade, which connects the promenade in Staro Sajmište (on the south, along Sava) and Zemunski Kej (on the north, Danube). The section of the Ušće promenade between the Branko's Bridge and the Hotel Jugoslavija has been renamed to the "Quay of the king Alexander I Karađorđević" on 29 June 2017.
The first planned bicycle path in Belgrade was built from Hotel Jugoslavija to Ušće. Designed by Mirko Radovanac, it was finished in 1979.
Motorsport circuit
Ušće was also the site of temporary street circuit where races were held since 1967. and were held almost every year afterwards. The track (in a 2.310 meter configuration) encompasses roads around the blok 15 (Ušće park), passing near the Ušće Tower, along the Branko's Bridge and then along the Sava bank, with its start-finish straight just across the eternal fire monument. The circuit is popularly known as "a circuit without right hand turn", although it has two slight right-handers, and it is famous for its high speed S-curve near restaurant "Ušće" and Museum of Contemporary Art, on Sava coast, which exist because the road had to navigate around the previously made Ranney collector. The first race was held on 30 April 1967. but the most remembered is the one held in 1969. (on a 4.350 configuration) when around 40.000 spectators flocked to see powerful Fiat Abarths in European Touring Car Challenge, and Formula Vee cars race on the streets. Young Niki Lauda participated in some of these races.
After poor management and maintenance of roads and circuit, and race cars reaching high speeds in 1990s and 2000s, FIA and drivers continuously criticised the safety of the track, lack of proper barriers and run-off areas. After many unfortunate events, and a fatality in 2005, the circuit was taken off the calendar and "Grand Prix of Belgrade" was subsequently replaced with a race held in Batajnica Air Base. Despite these events, the race on Ušće was held in 2008. The rich history of races held on Ušće have gained somewhat a "classic" status in Serbia and former Yugoslavia, with meany people's associations on hearing the word "Ušće" firstly reminding them about the race. Racing returned to the street circuit in 2016 and was added to the FIA Central European Zone Circuit Championship in 2017.
In September 2019 it was announced that the Ministry of the Interior's open garage under the Branko's Bridge will be adapted into the Museum of automobiles and Museum of the Ministry of the Interior. The project also includes plans for the adjoining tracks in Ušće to be used for revue races organized by the new museum.
Skatepark
After 20 years of skateboarders' activism to get one, the first skatepark in this part of Europe was opened on 28 September 2008. It is located in the Block 15, across the Ušće Tower and covers an area of , which made it the largest one in the region when opened. Originally, it had 15 elements but after the 2011 reconstruction, 6 new elements were added. Apart from the everyday use (entrance is free for everyone with skateboards, rollers and bicycles), local and international competitions are being held at the park.
In December 2018 it was announced that due to the construction of the aerial lift to Kalemegdan, the skatepark will be moved to another location, as the starting point of the route on this side of the river will be on the exact spot of the skatepark. Construction of the aerial lift caused various controversies and public protests, so in March 2019 city announced that the skate park will not only be preserved, as the route of the lift will be corrected, but that it is going to be expanded and modernized.
Belgrade Philharmonic
In 1961 an all-Yugoslav architectural design competition for the "Museum of the revolution of the nations and nationalities of Yugoslavia" was announced. It was to be built in Block 13, on the location between the modern Palace Serbia and Ušće Tower. Out of 29 design entries, the project of Vjenceslav Richter was chosen. Construction began only in 1978 and it was envisioned as a square with the alley, eternal flame and the monument to the President of Yugoslavia, Josip Broz Tito. The museum was planned to cover an area of but only the basement was built with the iron rods, for the projected reinforced concrete pillars, protruding above the ground ever since.
As the Belgrade Philharmonic Orchestra has no building of its own, and after the proposed locations for it on Vračar, near Gradić Pejton, or near the Hotel Jugoslavija, on the Danube river's bank, on 12 October 2016 the city announced an architectural design competition for the new building of the Belgrade Philharmonic on the location of the proposed museum and the surrounding terrain, in total. In January 2017, out of 34 design entries, the one by Dragan Marčetić and Milan Maksimović won the competition. The central hall will have 1,500 seats and the large stage. The smaller hall was designed to have 400 seats, as well as the outdoor hall. The building will cover an area of . A park and lookout are planned on the roof while coffee shops, restaurants and luxury boutiques will cover . The surrounding area will include bicycle paths, a small square, a fountain and an underground garage for 3,960 cars. The estimated value of the project is 30 million euros. In June 2018, Nikola Nikodijević, president of the Belgrade City hall, announced that construction should start in 2020.
Additionally, Marčetić and Maksimović also envisioned four 24-story skyscrapers west of the Palace Serbia, between the new Philharmonic and the YU Business Center shopping mall. The proposed skyscrapers will have a total floor area of and a two-level underground garage with .
Even though some deadlines were given and the project was used as the basis for the plan of the detailed regulation of the neighborhood, in August 2018 it was all dismissed. A new, international design competition was to be announced by the end of 2018. According to the new conditions, any high rise is forbidden and nothing can be higher than , which is the height of the Palace of Serbia, which is in the center of Block 13 since 1961. Not only that four skyscrapers will not be built, but even the existing gas station on that location will be removed and a new street will be cut through. One-floor underground garages next to the eastern and western wings of the Palace are also part of the new project. When it comes to the philharmonic itself, it also can't top the Palace's height nor block the view on it, and the architects will have to develop the building around the core ideas of the Arup Group, which was hired because of the acoustics of the future venue. None of the future objects will be fenced.
New competition was organized only in November 2020, and was to last to April 2021. Plans, permits and projects will be finished by 2022/2023, and the new building should be opened in 2026. Projected costs are €120 million, and will be financed by the funds from the Council of Europe Development Bank's credit. Design by the Amanda Levete's "AL_A" bureau was announced as the winner in April 2022, among 37 designs from 15 countries. Levete'с design, awarded $10,2 million, stipulates slightly larger hall, with 1,600 seats for audience and 400 for orchestra and choir, and a total of four large halls which would make the building - grand concert hall, recital hall, creative center and podium.
Gathering place
Due to its central location and wide-open space, Ušće has hosted some mass gatherings:
On November 19, 1988, Serbian Communist League's Central Committee chairman Slobodan Milošević addressed the crowd of more than 100,000 people as the main speaker at a mass rally that took place as integral part of "anti-bureaucratic revolution" and in essence served as Milošević's unofficial inauguration.
On March 11, 1991, Milošević's regime organized a large counter-rally in direct response to March 9th Protest two days earlier. The gathering was seen as the regime's attempt at showing that anti-regime protesters of the prior days were a lone, politically instrumentalized, and misguided group of destructive-minded youth. Milošević himself did not address the crowd, but some of the most prominent members of his Socialist Party of Serbia did. The crowd on hand mostly consisted of older generation members, many of whom were workers and pensioners bussed in for the occasion from other parts of Serbia.
On October 16, 2014, a military parade by the Serbian Army was held in front of more than 100,000 spectators for the 70th anniversary of the Soviet Red Army liberating city of Belgrade from Nazi Germany. The parade was also part of Russian president Vladimir Putin's official visit. Besides Putin who was the parade's centerpiece guest, the main stand featured Serbian dignitaries, president Tomislav Nikolić and prime minister Aleksandar Vučić as well as Republika Srpska president Milorad Dodik and Republika Srpska prime minister Željka Cvijanović.
Concerts and festivals
On July 3, 2004, the rock band Riblja Čorba and the Belgrade faction of the rock band Zabranjeno Pušenje headlined a day-long anniversary concert before 40,000 spectators that came out to celebrate Riblja Čorba's 25th and Zabranjeno Pušenje's 20th anniversary. However, the concert is best remembered by the fact that Zabranjeno Pušenje performance ended after about half an hour, as the band performed world music material they recorded as The No Smoking Orchestra, and the angry fans threw various objects on stage. Other acts that performed included Negative, Prljavi Inspektor Blaža i Kljunovi, Bjesovi, Alogia, Roze Poze, Abonos, Kraljevski Apartman, Đorđe David, and others.
On June 17, 2006, Serbian turbo-folk singer Ceca Ražnatović performed for more than three hours in front of the crowd of more than 100,000 people, in support of her Idealno loša album that got released the same day.
On July 14, 2007 The Rolling Stones finally played a concert in Serbia after several previous cancellations. The 2-hour show in front of approximately 70,000 people was part of their A Bigger Bang Tour.
On June 24, 2008, The Police played a concert in front of the crowd of 27,104 people as part of their reunion tour, with Counting Crows performing as the opening band.
Since 2008 Belgrade Beer Fest is held at Ušće every August.
On August 24, 2009, Madonna performed at Ušće in front of 39,713 people as part of the second leg of her Sticky & Sweet Tour.
On June 19 and June 21, 2010, Ušće Fest was held, with Bajaga i Instruktori and Vlado Georgiev playing on the 19th, and Billy Idol performing on the 21st. The festival was originally organized as a three-day event, but the scheduled performance by Goran Bregović and Severina on the second night, the 22nd, got cancelled due to stormy weather.
On June 25, 2011, Zdravko Čolić performed a 4-hour concert in front of 100,000 people.
On May 8, 2012, Metallica held a concert in front of approximately 30,000 people; they performed their 1991 eponymously titled album in its entirety. The concert featured Gojira and Machine Head as the opening bands.
From June 27 to June 30, 2012, Belgrade Calling four-day festival was held at Ušće. The first night of the festival featured Jessie J, Orbital, Example, Public Enemy, S.A.R.S. and Dubioza Kolektiv. The second evening of the festival featured Ozzy Osbourne (featuring Geezer Butler and Slash), The Cult, Black Label Society, Paradise Lost, Satyricon and Sanatorium. The third evening of the festival featured Faith No More, Public Image Limited, The Sisters of Mercy, The Darkness, The Horrors and Ugly Kid Joe. The last evening of the festival was free, and featured Serbian acts only: Straight Mickey & The Boys, Snakes, Sharks and Planes, Bjesovi, Svi Na Pod, Eva Braun, Goribor, Darkwood Dub, Kanda, Kodža i Nebojša and Block Out.
Depeche Mode performed at the park on 19 May 2013 during their Delta Machine Tour, in front of a sold-out crowd of 27,198 people. The band had previously been scheduled to perform at the park on 20 May 2009 as part of their Tour of the Universe, but the concert was cancelled due to singer Dave Gahan's severe bout of gastroenteritis.
On 15 June 2013, Jelena Karleuša performed a concert called Viva La Diva show in front of more than 40.000 people.
On 28 June 2013, Svetlana Ražnatović Ceca performed the biggest concert of her career which initiated her Poziv Tour in front of more than 100,000 people, promoting the album Poziv.
From 2014, CoffeeFest (Literary Serbian: Кофифест, Kofifest) a regional, annual coffee festival is held every last weekend in September on the Ušće street in Belgrade, Serbia across the Ušće Shopping Center.
On 17 June 2015, Robbie Williams performed at the park as part of his Let Me Entertain You Tour, in front of 33,000 people.
On 20 June 2015, Smak played a concert in front of 25,000 people.
From August 23 until August 25, 2019, Music Week three-day festival was held at Ušće. The first night of the festival featured Dragana Mirković, Tony Cetinski, Jelena Rozga, Saša Kovačević, and Dženan Lončarević. The second evening of the festival featured Lepa Brena, Aca Lukas, Nina Badrić, Hari Mata Hari, and Darko Lazić. The third evening of the festival featured Buba Corelli & Jala Brat, Maya Berović, Coby, Senidah, Sara Jo, Katarina Grujić, and Angellina.
References
Bibliography
Beograd – plan grada; M@gic M@p, 2006;
Neighborhoods of Belgrade
New Belgrade | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,914 |
Кизилташки лимани () језеро је лиманског порекла у старој делти реке Кубањ, на југу Таманског полуострва. Налази се на западу Краснодарске покрајине Русије, неких 25 км северозападно од града Анапе, а акваторија му је административно подељена између Анапског округа и Темрјучког рејона. Део је знатно пространије групације Кубањских лимана и један је од највећих лимана руског југа.
Лиман има неправилну округлу форму, максималне дужине у смеру запад-исток од 18,5 км и ширине у смеру север-југ од око 14 километара. Укупна површина лимана је 137 km². Све до почетка 20. века у лиман се уливао један од главних рукаваца реке Кубањ и самим тим вода у њему је углавном имала слатководни карактер. Променом корита Кубања и његовим преусмеравањем ка северу уплив слатке воде у језеро је знатно смањен, а данас је са изворним водотоком језеро повезано мањм рукавцем Старог Кубања. Од Црног мора на југу је одвојен уском Анапском косом. На северу је уском протоком повезан са Цокурским лиманом, на западу је Бугаски лиман, а на југоистоку Витјазевски лиман.
Дно језера прекривено је наслагама глине обогаћене високим концентрацијама водоник-сулфида због чега се користи у терапеутске сврхе као пелоид (лековито блато).
Види још
Краснодарски крај
Делта реке Кубањ
Кубањски лимани
Таманско полуострво
Референце
Спољашње везе
-{Лиманы Краснодарского края}-
-{Словарь географических названий Краснодарского края}-
-{Кизилташский лиман // Энциклопедический словарь Брокгауза и Ефрона : в 86 т. (82 т. и 4 доп.). — СПб., 1890—1907.}-
Језера у Русији
Краснодарски крај
Црно море
Слив Кубања | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,147 |
Sandfield Cemetery may refer to:
Sandfield Cemetery (Columbus, Mississippi), African-American cemetery founded in the late 19th-century.
Sandfield Cemetery (Richland County, South Carolina), Baptist cemetery founded c. 1772. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,003 |
Гай Кальвизий Сабин ():
Гай Кальвизий Сабин — консул в 39 году до н. э.
Гай Кальвизий Сабин — консул в 4 году до н. э.
Гай Кальвизий Сабин — консул в 26 году | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,802 |
Secastilla is een gemeente in de Spaanse provincie Huesca in de regio Aragón met een oppervlakte van 47 km². In 2004 telde Secastilla 147 inwoners.
Demografische ontwikkeling
Bron: INE, 1857-2011: volkstellingen
Opm.: In 1857 werden de gemeenten Puidecinca, Ubiergo en Bolturina aangehecht
Gemeente in Huesca | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,568 |
{"url":"https:\/\/socratic.org\/questions\/what-is-the-size-of-boron-in-relation-to-oxygen-and-aluminum","text":"# What is the size of boron in relation to oxygen and aluminum?\n\nJun 10, 2017\n\nAtomic size decreases ACROSS a Period.........\n\n#### Explanation:\n\nAtomic size decreases ACROSS a Period, from left to right as we face the table. And INCREASES down a Group, a column of the Periodic Table. These are very important Periodic Trends that you should try to understand in detail, as they relate directly to atomic structure.\n\nSo $\\text{2nd row}$ boron is SMALLER than $\\text{third row}$ aluminum.\n\nBut the $\\text{Group 16}$ oxygen atom is SMALLER than the $\\text{Group 13}$ boron atom.\n\nI am unwilling to leave it here, in that as physical scientists, we should always interrogate data, and see if the numbers support our (maybe witless) argument.\n\nThis site reports that the atomic radii of $B , O ,$ and $A l$ are 125 pm, 85 pm, and 160 pm respectively ($1 \\cdot \\text{pm} = {10}^{-} 12 \\cdot m$). The numbers at least support our argument.\n\nNote that in an exam you would never be expected to remember these data; however, you would be expected to interpret them.","date":"2022-09-27 09:11:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 7, \"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\": 0.7164670825004578, \"perplexity\": 876.8699755054091}, \"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-2022-40\/segments\/1664030334992.20\/warc\/CC-MAIN-20220927064738-20220927094738-00790.warc.gz\"}"} | null | null |
Das Rathaus von Artà (katalanisch: Ajuntament d'Artà) ist das Rathaus der spanischen Gemeinde Artà im nordöstlichen Teil der Mittelmeerinsel Mallorca.
Das Rathaus befindet sich im Zentrum Artàs am Plaça de l'Ajuntament.
Architektur und Geschichte
Das Gebäude wurde im Jahr 1941 als Ersatz für ein älteres Rathaus errichtet. Der Plan für einen Neubau stammte bereits aus der Zeit der Zweiten Republik, zwischen 1931 und 1936. Bedingt durch den Spanischen Bürgerkrieg wurde das Vorhaben jedoch zunächst wieder eingestellt und der Bau erst nach Ende des Bürgerkriegs weitergeführt. Das Rathaus wurde gemeinsam mit dem davor befindlichen Platz 1941 eingeweiht.
Nach Ende der Franco-Diktatur wurden Symbole der Diktatur vom Gebäude entfernt.
Der dreigeschossige Bau ist symmetrisch ausgeführt. Im Untergeschoss befand sich eine Arrestzelle. Im ersten Obergeschoss sind Büros, darunter das Büro des Bürgermeisters sowie der Plenarsaal untergebracht. Im Dachgeschoss befindet sich ein Archiv.
Weblinks
Informationen zum Rathaus Artà
Bauwerk in Artà
Artà
Erbaut in den 1940er Jahren
Standort der Kommunalverwaltung | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,587 |
\section*{\refname}}
\begin{document}
\title{Bayesian inflationary reconstructions from \textit{Planck}{} 2018 data}
\author{Will J.\ Handley}
\email[]{wh260@mrao.cam.ac.uk}
\affiliation{Astrophysics Group, Cavendish Laboratory, J.J.Thomson Avenue, Cambridge, CB3 0HE, UK}
\affiliation{Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK}
\affiliation{Gonville \& Caius College, Trinity Street, Cambridge, CB2 1TA, UK}
\author{Anthony N.\ Lasenby}
\email[]{a.n.lasenby@mrao.cam.ac.uk}
\affiliation{Astrophysics Group, Cavendish Laboratory, J.J.Thomson Avenue, Cambridge, CB3 0HE, UK}
\affiliation{Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK}
\author{Hiranya V.\ Peiris}
\email[]{h.peiris@ucl.ac.uk}
\affiliation{The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, Stockholm, SE-106 91, Sweden}
\affiliation{Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, UK}
\author{Michael P.\ Hobson}
\email[]{mph@mrao.cam.ac.uk}
\affiliation{Astrophysics Group, Cavendish Laboratory, J.J.Thomson Avenue, Cambridge, CB3 0HE, UK}
\date{\today}
\Crefname{equation}{Equation}{Equations}
\crefname{equation}{Eq.}{Eqs.}
\Crefname{figure}{Figure}{Figures}
\crefname{figure}{Fig.}{Figs.}
\Crefname{table}{Table}{Tables}
\crefname{table}{Tab.}{Tabs.}
\Crefname{section}{Section}{Sections}
\crefname{section}{Sec.}{Secs.}
\begin{abstract}
We present three non-parametric Bayesian primordial reconstructions using \textit{Planck}{} 2018 polarization data: linear spline primordial power spectrum reconstructions, cubic spline inflationary potential reconstructions and sharp-featured primordial power spectrum reconstructions. All three methods conditionally show hints of an oscillatory feature in the primordial power spectrum in the multipole range $\ell\sim20$ to $\ell\sim50$, which is to some extent preserved upon marginalization. We find no evidence for deviations from a pure power law across a broad observable window ($50\lesssim\ell\lesssim2000$), but find that parameterizations are preferred which are able to account for lack of resolution at large angular scales due to cosmic variance, and at small angular scales due to \textit{Planck}{} instrument noise. Furthermore, the late-time cosmological parameters are unperturbed by these extensions to the primordial power spectrum. This work is intended to provide a background and give more details of the Bayesian primordial reconstruction work found in the \textit{Planck}{} 2018 papers.
\end{abstract}
\pacs{}
\maketitle
\section{Introduction\label{sec:introduction}}
The final release of \textit{Planck}{} satellite data~\cite{planck_legacy,planck_likelihood,planck_parameters,planck_inflation} provides an unprecedented window onto the cosmic microwave background (CMB). These high-resolution CMB anisotropy data give constraints on the state of the Universe in its earliest observable stage. Assuming a theory of inflation, the primordial power spectrum of curvature perturbations provides an indirect probe of ultra-high energy physics.
This paper focuses on non-parametric reconstructions of primordial physics. The aim of such analyses is to provide information on quantities and functions of interest that are arguably model-independent. While unambiguous scientific detections will only ever result from a consideration of specific, physically motivated models, results from reconstructions such as these can be used to inform and guide observational and theoretical cosmology, providing insight and evidence for interesting features not clearly visible in the data when using standard modeling assumptions.
Throughout we adopt a fully Bayesian framework, treating our non-parametric reconstruction functions using priors, posteriors and evidences to marginalize out factors that are irrelevant to physical quantities of interest. We reconstruct both the inflationary potential and the primordial power spectrum directly using spline and feature-based reconstructions, in a manner related but not identical to the existing literature~\citep{Vazquez_primordial,Aslanyan_primordial,Core_inflation,planck_inflation_2013,planck_inflation_2015,Hee_dark_energy,Vazquez_dark_energy,marius_reionisation,Malak_clusters,Bucher,2010PhRvD..81b3518D, 2010PhRvD..82d3513D, 2011PhRvD..84f3515D, 2015PhRvD..91f3514M, 2017PhRvD..96h3526O, 2018PhRvD..98d3518O}.
In \cref{sec:background} we review the relevant background theory in primordial cosmology, Bayesian inference, non-parametric reconstruction and CMB data. \Cref{sec:PPSR} reconstructs the primordial power spectrum directly using a linear interpolating spline. \Cref{sec:vphi_reconstruction} takes the analysis one step back and reconstructs the inflationary potential using a cubic spline, treating the primordial power spectrum as a derived quantity. \Cref{sec:features} works with a parameterization that is more suited for reconstructing sharp features in the primordial power spectrum as a complementary approach to that of \cref{sec:PPSR}. \Cref{sec:conclusion} draws conclusions from all three analyses.
\section{Background\label{sec:background}}
\subsection{Primordial cosmology\label{sec:background_primordial_cosmology}}
We begin by summarizing the background theory and establish notation. For a more detailed discussion of inflationary cosmology and perturbation theory, we recommend \citet{mukhanov} or \citet{baumann,baumann_2}.
The evolution equations for a spatially homogeneous, isotropic and flat universe filled with a scalar field $\phi$ with arbitrary potential $V(\phi)$ are
\begin{gather}
\ddot{\phi} + 3 H \dot{\phi} + \frac{\d{V}}{\d{\phi}} = 0,
\label{eqn:klein_gordon}\\
H^2 = \frac{1}{3M_\p^2}\left( \frac{1}{2}\dot{\phi}^2 + V(\phi) \right),
\label{eqn:Friedmann}
\end{gather}
where $H=\frac{\dot{a}}{a}$ is the Hubble parameter, $a$ is the scale factor of the Universe, dots denote derivatives with respect to cosmic time $\dot{f}\equiv \frac{\d{f}}{\d{t}}$ and $M_\p$ is the reduced Planck mass. For most potentials $V(\phi)$, solutions to \cref{eqn:klein_gordon,eqn:Friedmann} rapidly converge on the attractor {\em slow roll\/} state, satisfying $\dot{\phi}^2\ll V(\phi)$.
The evolution equations for the Fourier $k$-components of the gauge-invariant comoving curvature $\mathcal{R}$ and tensor $\mathcal{T}$ perturbations are
\begin{align}
\mathcal{R}_k'' + 2\frac{z'}{z} \mathcal{R}_k' + k^2 \mathcal{R}_k &=0,
\label{eqn:mukhanov_sazaki_scalar}\\
\mathcal{T}_k'' + 2\frac{a'}{a} \mathcal{T}_k' + k^2 \mathcal{T}_k &=0,
\label{eqn:mukhanov_sazaki_tensor}\\
z = \frac{a \dot{\phi}}{H}, \qquad \eta = \int \frac{\d{t}}{a}&,
\end{align}
where primes denote derivatives with respect to conformal time $f' \equiv \frac{\d{f}}{\d{\eta}}$. These equations have the property that in-horizon solutions ($k\gg aH$) oscillate with time-varying amplitude and frequency, whilst out-of-horizon solutions ($k\ll aH$) freeze out.
The dimensionless primordial power spectra of these perturbations are defined as
\begin{equation}
\mathcal{P}_{\mathcal{X}}(k) = \lim_{aH \gg k}\frac{k^3}{2\pi^2}\left|\mathcal{X}_k\right|^2, \quad \mathcal{X}\in\{\mathcal{R},\mathcal{T}\}.
\label{eqn:power_spectrum}
\end{equation}
Initial conditions for the background \cref{eqn:klein_gordon,eqn:Friedmann} may be set using the slow roll approximation:
\begin{equation}
H = \sqrt{\frac{V(\phi_\mathrm{init})}{3M_\p^2}}, \quad
\dot{\phi} = -\frac{V^{\prime}(\phi_\mathrm{init})}{3H},
\label{eqn:background_sr}
\end{equation}
where $V^\prime$ denotes the derivative of $V$ with respect to $\phi$.
Whilst solutions set with these initial conditions do not lie precisely on the attractor state, they rapidly converge on it. Providing that $\phi_\mathrm{init}$ is chosen self-consistently~\citep{Hiranya_potential_1,Hiranya_potential_2,Hiranya_potential_3} with enough additional evolution so that any transient effects are lost, these initial conditions are equivalent to choosing the background solution to be the attractor.
For the perturbation \cref{eqn:mukhanov_sazaki_scalar,eqn:mukhanov_sazaki_tensor}, Bunch-Davies initial conditions are chosen such that the Mukhanov variables match onto the de-Sitter vacuum solutions
\begin{equation}
\mathcal{R}_k = \frac{1}{z\sqrt{2k}}e^{-ik\eta}, \quad \mathcal{T}_k = \frac{1}{a\sqrt{2k}}e^{-ik\eta}.
\end{equation}
Providing that the $k$-mode lies well within the horizon ($k\gg aH$), this is the canonical choice for initializing the perturbation spectrum, although other vacua are available~\cite{Lim,NovelQuantum,PowerSpectrum,Fruzsina}.
Whilst \cref{eqn:klein_gordon,eqn:Friedmann,eqn:mukhanov_sazaki_scalar,eqn:mukhanov_sazaki_tensor} take their simplest form using cosmic and conformal time as variables, for numerical stability it is more prudent to choose a time-like parameter which does not saturate during inflation, such as cosmic time $t$, or the number of $e$-folds $N=\log(a)$. We choose the logarithmic comoving horizon $\log(a H)$ as the independent variable for our analyses. In this form \cref{eqn:klein_gordon,eqn:Friedmann,eqn:mukhanov_sazaki_scalar,eqn:mukhanov_sazaki_tensor} become complicated, so to avoid typographical errors we generate Fortran source code using the Maple~\cite{Maple} computer algebra package. The numerical integration of all differential equations was performed using the NAG library~\cite{NAG}.
It should also be noted that \cref{eqn:mukhanov_sazaki_scalar,eqn:mukhanov_sazaki_tensor} are usually phrased in terms of the Mukhanov variables $v =z\mathcal{R}$ and $h=a\mathcal{T}$, but $\mathcal{R}$ and $\mathcal{T}$ prove to be more numerically stable as they have the attractive property that they explicitly freeze out.
\subsection{Bayesian statistics\label{sec:background_bayesian_statistics}}
Once the primordial power spectra $\mathcal{P}_{\mathcal{R},\mathcal{T}}(k)$ have been determined, these form the initial conditions for Boltzmann codes~\cite{camb,class}. For a universe described by a cosmological model $M$ with corresponding late-time parameters $\Theta_\mathrm{c}$ and primordial power spectra $\mathcal{P}$, a Boltzmann code computes CMB power spectra $C_\mathrm{\ell}$ in both temperature and polarization. These CMB power spectra may then be fed into cosmological likelihood codes~\cite{planck_likelihood}, which typically depend on additional nuisance parameters $\Theta_\mathrm{n}$ associated with the experiment. The end result is a likelihood $P(D|\Theta,M)$ of the parameters $\Theta=(\mathcal{P}(k),\Theta_\mathrm{c},\Theta_\mathrm{n})$ given CMB data $D$, and a cosmological model $M$.
We may formally invert the conditioning on $\theta$ in the likelihood using Bayes theorem
\begin{align}
P(\Theta|D,M) &= \frac{P(D|\Theta,M)P(\Theta,M)}{P(D|M)},
\label{eqn:parameter_estimation}\\
P(D|M) &= \int P(D|\Theta,M)P(\Theta|M)\d{\Theta},
\label{eqn:evidence}
\end{align}
where the first expression above should be read as ``{\em posterior\/} is {\em likelihood\/} times {\em prior\/} over {\em evidence}'', and the second expression indicates that the evidence is the normalizing constant of \cref{eqn:parameter_estimation}, and is a multidimensional marginalization of the likelihood over the prior. The evidence may also be used in a Bayesian model comparison, to assess the relative merits of a set of competing models $\{M_i\}$
\begin{align}
P(M_i|D) &= \frac{P(D|M_i)P(M_i)}{P(D)},
\label{eqn:model_comparison}\\
P(D) &= \sum_i P(D|M_i)P(M_i).
\end{align}
In the event of uniform priors over models, the evidence establishes the relative probability weighting to give to models describing the same data $D$.
Throughout this work, we use a modified version of \texttt{CAMB}~\cite{camb} to compute $C_\ell$ power spectra and \texttt{CosmoChord}~\cite{cosmochord} (a modified version of \texttt{CosmoMC}~\cite{cosmomc,cosmomc_fs}) to interface the likelihoods. To sample the posterior and compute evidences we make use of the nested sampling~\cite{skilling2006} algorithm \texttt{PolyChord}~\cite{PolyChord0,PolyChord1}. The default Metropolis--Hastings sampler in~\texttt{CosmoMC}{} is insufficient both due to the complexity of the posteriors that must be navigated, and the requirement of evidence computation. Furthermore, \texttt{PolyChord}{} is required in place of the previous nested sampling algorithm \texttt{MultiNest}~\cite{MultiNest0,MultiNest1,MultiNest2} due to the high dimensionality of the full \textit{Planck}{} likelihood with nuisance parameters. As an added bonus, \texttt{PolyChord}{} has the ability to exploit the fast-slow cosmological hierarchy~\cite{cosmomc_fs}, which greatly speeds up the sampling. Most importantly all parameters associated with the primordial power spectrum are ``semi-slow'', given that one does not need to recompute transfer functions upon changing the primordial power spectrum.
\subsection{Functional inference\label{sec:background_functional_inference}}
For our reconstructions, the quantities of interest are functions $f(k;\Theta_f)$ of wavenumber $k$, parameterized by a set of parameters $\Theta_f$, which presents a challenge in both plotting and quantifying our results.
We utilize two related techniques to plot the posterior of a function $f(k;\Theta_f)$. First, we can generate equally-weighted samples of $\Theta_f$, and therefore of the function $f$, and plot each sample as a curve on the $(k,f)$ plane. In general, we simultaneously plot prior samples in red, and posterior samples in black. An example of such a plot can be found in the upper-left panel of \cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps}. For the second type of plot, we first compute the marginalized posterior distribution $P(f|k)$ of the dependent variable $f$ conditioned on the independent variable $k$ using Gaussian kernel density estimation. The iso-probability credibility intervals are then plotted in the $(k,f)$ plane, with their mass converted to $\sigma$-values via an inverse error function transformation. An example of this kind of plot can be seen in the upper-right panel of \cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps}. The code for producing such plots is published in Ref.~\cite{fgivenx}.
To quantify the constraining power of a given reconstruction, we use the conditional Kullback-Leibler (KL) divergence~\cite{Kullback:1951} as exemplified by~\citet{Hee_dark_energy}. For two distributions $P(x)$ and $Q(x)$, the KL divergence is defined as
\begin{equation}
D_{KL}(P|Q) = \int \ln \left[ \frac{P(x)}{Q(x)} \right] P(x) \d{x},
\end{equation}
and may be interpreted as the information gain in moving from a prior $Q$ to a posterior $P$~\citep{Hoyosa:2004,Verde:2013,Grandis:2016,Raveri2016}. For our reconstructions, we compute the KL divergence from prior to posterior for each distribution $P(f|k)$ conditioned on $k$. An example of such a plot can be found in the lower-right panel of \cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps}.
Throughout this work, plots use an approximate correspondence between wavenumber $k$ and multipole moment $\ell$ via the Limber approximation $\ell\approx k/D_\mathrm{A}$, where $D_\mathrm{A}=r_*/\theta_*$ is the \textit{Planck}{} 2018 best-fit comoving angular distance to recombination at $r_*$.
\subsection{Non-parametric reconstructions\label{sec:background_non_parametric}}
Throughout this work, we explore various non-parametric functional forms for either the primordial power spectrum or the inflationary potential. ``Non-parametric'' is a slightly misleading terminology, as in general such reconstructions choose a function with a very large number of additional parameters. We prefer the terminology free-form~\cite{planck_inflation,Malak_clusters}, flexible~\cite{marius_reionisation} or adaptive~\cite{Hee_dark_energy,HthreeL}. The principle behind this is that the parameterization should have enough freedom to reconstruct any reasonable underlying function, independent of any underlying physical model.
For example, in this paper we work with variations on the {\em linear spline}, defined by parameters $\Theta_f$, producing a mapping from the independent variable $x$ to the dependent variable $y$ thus
\begin{align}
\mathrm{Lin}(x;\Theta_f) &=
\sum\limits_{i=1}^N\tfrac{y_{i}(x_{i+1}-x) + y_{i+1}(x-x_{i})}{x_{i}-x_{i+1}}
\left[x_{i} < x \le x_{i+1}\right],
\nonumber\\
\Theta_f &= (x_1,\cdots,x_N,y_1,\cdots,y_N).
\label{eqn:linear_spline}
\end{align}
Here we have used a compact notation for denoting piecewise functions espoused by~\citet{ConcreteMathematics} whereby $[R]$ is a logical truth function, yielding $1$ if the relation $R$ is true, and $0$ if false.
For consistency, we interpret the case $N=1$ as having a constant value of $y_1$ for all $x$.
In a Bayesian approach, one treats the additional degrees of freedom $\Theta_f$ of the non-parametric function as parameters in a posterior distribution, which one marginalizes out in order to obtain model-independent reconstructions. Typically there is a degree of choice as to how many parameters $N$ to use, and a penalty is applied for larger $N$ to avoid over-parameterization and noise fitting. In this work we treat $N$ in a Bayesian sense as well. Each reconstruction with a given number of parameters $N$ is treated as an independent model. We can then marginalize over the number of models using the Bayesian evidence.
\subsection{\textit{Planck}{} data and cosmology\label{sec:background_data}}
In this paper in almost all cases we focus our efforts on using the pure \textit{Planck}{} 2018 polarization data baseline, referred to in~\cite{planck_legacy,planck_likelihood,planck_parameters,planck_inflation} as TT,TE,EE+lowE+lensing.
Throughout we use a flat cold-dark-matter with dark energy ($\Lambda$CDM) late-time cosmology; therefore there are four associated cosmological parameters which in the default \texttt{CosmoMC}{} basis take the form
\begin{equation}
\Theta_\mathrm{c} = (\Omega_b h^2, \Omega_c h^2, 100\theta_{MC}, \tau).
\end{equation}
Additionally there are $21$ nuisance parameters associated with Galactic foregrounds and the \textit{Planck}{} instrumentation:
\begin{align}
\Theta_\mathrm{n} = (&y_{\rm cal}, A^{CIB}_{217}, \xi^{tSZ-CIB}, A^{tSZ}_{143}, A^{PS}_{100}, A^{PS}_{143}, A^{PS}_{143\times217}, \nonumber\\
&A^{PS}_{217}, A^{kSZ}, A^{{\rm dust}TT}_{100}, A^{{\rm dust}TT}_{143}, A^{{\rm dust}TT}_{143\times217}, A^{{\rm dust}TT}_{217}, \nonumber\\
&A^{{\rm dust}TE}_{100}, A^{{\rm dust}TE}_{100\times143}, A^{{\rm dust}TE}_{100\times217}, A^{{\rm dust}TE}_{143}, A^{{\rm dust}TE}_{143\times217}, \nonumber\\
&A^{{\rm dust}TE}_{217}, c_{100}, c_{217}).
\end{align}
We use the \textit{Planck}{} \texttt{PolyChord}{} \texttt{CosmoMC}{} defaults as prior widths for all of these (indicated in \cref{tab:default_prior}), but as they are common to all models considered and sufficiently wide to encompass the entire posterior bulk, any prior effects from these parameters have been shown theoretically~\cite{2007MNRAS.378...72T} and in practice~\cite{2019PhRvD.100d3504H} to cancel out.
\subsection{Sampling strategy}
Throughout this paper, we sample over the full parameter space $\Theta=(\Theta_\mathrm{f},\Theta_\mathrm{c},\Theta_\mathrm{n})$ of reconstruction, cosmological and nuisance parameters. The resulting posteriors are in general multimodal with complicated degeneracies between many parameters, particularly when there are a large number of reconstruction parameters $\Theta_\mathrm{f}$. In all cases, plots in this paper have the unmentioned parameters implicitly marginalized out. Marginalizing the likelihood over the prior in order to compute evidences is in general even more challenging. Nested sampling is ideally suited to performing such tasks, with \texttt{PolyChord}{} providing the cutting-edge of such technology in a cosmological context~\cite{PolyChord0,PolyChord1}, proving to be essential for sampling over these complicated parameter spaces with up to $\sim\mathcal{O}(50)$ dimensions with fast-slow parameter hierarchies.
\begin{table}[th]
\begin{tabular}{llll}
Parameters& Prior type& Prior parameters & Speed \\
\hline
$\Omega_b h^2$ & uniform & $[0.019,0.025]$ & slow\\
$\Omega_c h^2$ & uniform & $[0.095,0.145]$ & slow\\
$100\theta_{MC}$ & uniform & $[1.03,1.05]$ & slow\\
$\tau$ & uniform & $[0.01,0.4]$ & slow\\
\hline
$y_{\rm cal}$ & Gaussian & $1\pm 0.0025$ & semi-slow\\
$A^{CIB}_{217}$ & uniform & $[0, 200]$ & fast\\
$\xi^{tSZ-CIB}$ & uniform & $[0, 1]$ & fast\\
$A^{tSZ}_{143}$ & uniform & $[0, 10]$ & fast\\
$A^{PS}_{100}$ & uniform & $[0, 400]$ & fast\\
$A^{PS}_{143}$ & uniform & $[0, 400]$ & fast\\
$A^{PS}_{143\times217}$ & uniform & $[0, 400]$ & fast\\
$A^{PS}_{217}$ & uniform & $[0, 400]$ & fast\\
$A^{kSZ}$ & uniform & $[0, 10]$ & fast\\
$A^{{\rm dust}TT}_{100}$ & Gaussian & $8.6\pm 2$ & fast\\
$A^{{\rm dust}TT}_{143}$ & Gaussian & $10.6\pm 2$ & fast\\
$A^{{\rm dust}TT}_{143\times217}$ & Gaussian & $23.5\pm 8.5$ & fast\\
$A^{{\rm dust}TT}_{217}$ & Gaussian & $91.9\pm 20$ & fast\\
$A^{{\rm dust}TE}_{100}$ & Gaussian & $0.13\pm 0.042$ & fast\\
$A^{{\rm dust}TE}_{100\times143}$ & Gaussian & $0.13\pm 0.036$ & fast\\
$A^{{\rm dust}TE}_{100\times217}$ & Gaussian & $0.46\pm 0.09$ & fast\\
$A^{{\rm dust}TE}_{143}$ & Gaussian & $0.207\pm 0.072$ & fast\\
$A^{{\rm dust}TE}_{143\times217}$ & Gaussian & $0.69\pm 0.09$ & fast\\
$A^{{\rm dust}TE}_{217}$ & Gaussian & $1.938\pm 0.54$ & fast\\
$c_{100}$ & Gaussian & $1.0002\pm 0.0007$ & fast\\
$c_{217}$ & Gaussian & $0.99805\pm 0.00065$ & fast\\
$A^{kSZ} + 1.6 A^{tSZ}_{143}$ & Gaussian & $9.5\pm 3$ & fast\\
\end{tabular}
\caption{The prior distributions on late-time cosmological parameters and \textit{Planck}{} nuisance parameters for all analyses. The parameters of each Gaussian distribution are defined as $[\mu, \sigma]$, and the above distributions combine to make a truncated Gaussian distribution on the nuisance parameters $A^{kSZ}$ and $A^{tSZ}_{143}$. The nuisance priors are the default ones in \texttt{CosmoMC}{}, whilst the cosmological priors are narrowed to speed up sampling, but remain sufficiently wide to effectively include the entire posterior mass. Also indicated is each parameter's speed with respect to the \texttt{CosmoMC}{} fast-slow hierarchy.\label{tab:default_prior}}
\end{table}
\clearpage
\newpage
\section{Primordial power spectrum reconstruction\label{sec:PPSR}}
\FloatBarrier
\begin{figure}[t]
\includegraphics{knots}
\caption{We parameterize the primordial power spectrum reconstruction via a linear interpolating spline in the $(\ln k, \ln \mathcal{P})$ plane with $N$ spline locations $(k_1,\mathcal{P}_1),\cdots(k_N,\mathcal{P}_N)$. The outermost $k$-locations are fixed, with the inner locations constrained by $k_1<\cdots<k_N$, and the entire spline constrained within the box indicated by the dashed line.\label{fig:pps_parametrisation}}
\end{figure}
\begin{table}[t]
\begin{tabular}{lll}
Parameters& Prior type& Prior range \\
\hline
$N$ & discrete uniform & $[1, 9]$ \\
$\mathcal{P}_1,\cdots,\mathcal{P}_N$ & $\log$-uniform & $10^{-10}[e^2,e^4]$\\
$k_2<\cdots<k_{N-1}$ & sorted $\log$-uniform & $[10^{-4},10^{-0.3}]$ \\
\end{tabular}
\caption{The prior distributions on early-time cosmological parameters for the primordial power spectrum reconstruction.\label{tab:pps_prior}}
\end{table}
Traditionally in a $\Lambda$CDM cosmology, the primordial power spectrum $\mathcal{P}_\mathcal{R}(k)$ is modeled by a two-parameter function with an amplitude $A_\mathrm{s}$ and spectral index $n_\mathrm{s}-1$
\begin{equation}
\ln \mathcal{P}_\mathcal{R}(k) = \ln A_\mathrm{s} + (n_\mathrm{s} -1) \ln\left( \frac{k}{k_*} \right),
\label{eqn:lcdm_parametrisation}
\end{equation}
i.e.\ a straight line in the $(\ln k,\ln\mathcal{P})$ plane. The tensor spectrum $\mathcal{P}_\mathcal{T}(k)$ may be parameterized by its own independent amplitude $A_\mathrm{t}$ and index $n_\mathrm{t}$, or via the tensor to scalar ratio $r=A_\mathrm{t}/A_\mathrm{s}$ and slow roll inflation consistency condition $n_\mathrm{t}=-r/8$~\cite{baumann}. Extensions to parameterization~\eqref{eqn:lcdm_parametrisation} can be made by adding quadratic (running) and cubic (running of running) terms, but no evidence is found that these are required to describe the primordial power spectrum in the $k$-window which \textit{Planck}{} probes.
Extending \cref{eqn:lcdm_parametrisation} with runnings of the spectral index creates a stiff parameterization, with no ability to account for sharper features, or large deviations at low or high-$k$. For our first primordial power spectrum reconstruction, we therefore parameterize as a logarithmic spline
\begin{align}
\ln10^{10}\mathcal{P}_\mathcal{R}(k) =& \mathrm{Lin}(\log_{10} k;\Theta_\mathcal{P})\nonumber\\
\Theta_\mathcal{P} =& (\log_{10} k_1,\cdots,\log_{10} k_N,\nonumber\\
&\ln 10^{10}\mathcal{P}_1,\cdots,\ln 10^{10} \mathcal{P}_N).
\label{eqn:pps_parametrisation}
\end{align}
This represents a spline (\cref{eqn:linear_spline}) that is linear in the $(\ln k,\ln\mathcal{P})$ plane as shown in \cref{fig:pps_parametrisation}. For the tensor power spectrum we have analyzed the cases where $r$ is allowed to vary as a parameter, and also when $\mathcal{P}_\mathcal{T}(k)$ is given its own independent linear spline. Unsurprisingly, given that \textit{Planck}{} measured an $r$ consistent with $0$, the addition of a tensor power spectrum makes no difference to the scalar reconstructions. For simplicity we assume $r=0$ for the remainder of this section.
This technique has a history of being successfully applied to the primordial power spectrum~\citep{Vazquez_primordial,Aslanyan_primordial,Core_inflation,planck_inflation_2013,planck_inflation_2015}, but has also been applied to dark energy equation of state by \citet{Hee_dark_energy} and \citet{Vazquez_dark_energy}, to the cosmic reionisation history by \citet{marius_reionisation} and to galaxy cluster profiles by \citet{Malak_clusters}.
Our work differs from previous primordial power spectrum reconstructions in both the data we use, the styling of the priors, and in the application of more modern inference tools such as functional posterior plotting~\cite{fgivenx}, conditional Kullback-Leibler divergences~\cite{Hee_dark_energy} and \texttt{PolyChord}{}. The technique was applied to primordial power spectrum reconstruction from CORE simulated data in Section 6 of~\cite{Core_inflation}, where it was shown that this approach accurately reconstructs complicated injected features (or lack thereof).
\subsection*{Priors}
For priors on the vertical spline location parameters, we choose them to be independently uniform in $2<\log 10^{10}\mathcal{P}<4$. This spans an almost maximally wide range, increasing their width further has little effect due to \texttt{CosmoMC}{} discarding unphysically normalized spectra.
For priors on the horizontal spline location parameters, we choose the outermost knots to be fixed at $10^{-4}$ and $10^{-0.3}$. This corresponds roughly to a $C_\ell$ multipole range $1\lesssim\ell\lesssim7000$, which fully encompasses the CMB window that \textit{Planck}{} observes. For the remaining horizontal knots, we choose a prior which distributes the parameters logarithmically within this range, such that $k_2<\cdots<k_{N-1}$. This sorting procedure breaks the $(N-2)!$ implicit switching degeneracy, and is also termed a {\em forced identifiability prior}~\cite{PolyChord1,THE,forcedidentifiability}.
To implement this sorted prior in the context of nested sampling, we need to define the transformation from the unit hypercube to the physical space. Coordinates in the unit hypercube $x_1,\cdots,x_N$, can be transformed to coordinates in the physical space $\theta_1,\cdots,\theta_N$, such that they are distributed uniformly in $[\theta_{\min{}},\theta_{\max{}}]$ and sorted so that $\theta_1<\cdots<\theta_N$ via the following reversed recurrence relation
\begin{equation}
\theta_{n} = \theta_{\min{}} + (\theta_{n+1}-\theta_{\min{}}) x_{n}^{1/n},
\qquad
\theta_{N+1} = \theta_{\max{}},
\end{equation}
which is equivalent to saying that $\theta_n$ is marginally distributed as the largest of $n$ uniformly distributed variables within $[\theta_{\min{}},\theta_{n+1}]$.
Another method for breaking the switching degeneracy is to exclude the region of the parameter space which does not satisfy the sorting criterion. This becomes exponentially small as more knots are added, which makes the initial sampling from the prior more challenging. It is also more in keeping with the nested sampling methodology to explicitly transform the full hypercube onto the space of interest.
In our case, given that $k_1$ and $k_N$ are fixed, for $N\ge4$, we sort the $N-2$ inner logarithmic coordinates $\log_{10} k_2<\cdots<\log_{10} k_{N-1}$.
We perform the reconstruction for $N=1,\cdots,9$ and then marginalize using Bayesian evidences with an implicit equal weighting for each $N$. This is equivalent to sampling from a full joint posterior with a uniform prior on $N$, and could alternatively be accomplished using the method described in~\citet{HthreeL}. Our priors on the reconstruction parameters are summarized in \cref{tab:pps_prior}.
\subsection*{Results}
We show results for our primordial power spectrum reconstruction using \textit{Planck}{} 2018 TT,TE,EE+lowE+lensing data in \cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid,fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps_both_grid,fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps}.
\Cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid,fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps_both_grid} show the prior and posterior conditioned on each value of $N$. The case $N=1$ corresponds to a scale-invariant spectrum, whilst $N=2$ is equivalent (up to a small difference in prior) to the standard $\Lambda$CDM parameterization. As further knots are added, the reconstruction accounts for cosmic variance at low-$k$, and instrument noise at high-$k$. Furthermore, for large $N$, in a fraction of the samples there is a visibly clear oscillation characterized by a rise in power at $\ell\sim50$, and a dip in power at $20<\ell<30$, as well as an overall suppression of power at low-$k$. For lower values of $\ell$, cosmic variance sets in, and few conclusions can be drawn from sampling differences between runs of different $N$ at these values.
To determine the statistical significance of these features, one should consider the Bayesian evidence, as indicated in the lower-left panel of \cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps}.
The first observation from \cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps} is that the $N=1$ scale-invariant power spectrum is completely ruled out, with a logarithmic difference of $\ln\mathcal{B}_{N=1}^{N=2}\sim\mathcal{O}(33)$. This represents overwhelming evidence for a tilted power spectrum, one of the key predictions of the theory inflation. A gambler could get odds of a quintillion to one against scale-invariance vs.\ $\Lambda$CDM.
The second observation is that the evidence for $N=3$ is greater than $N=2$, namely a model that is able to account for cosmic variance at low-$k$ and instrument noise at high-$k$ is, in a Bayesian sense, preferred to the simpler $\Lambda$CDM parameterization. Up until \textit{Planck}{} 2018, the data had not been quite powerful enough for us to define the window that we observe in the primordial power spectrum in this Bayesian sense.
The third observation is that whilst $N=3$ is maximal in evidence, in fact $N=4,5,6$ are competitive, and $N=7,8,9$ are far from ruled out. With this lack of knowledge, the correct Bayesian approach is to marginalize over all models, using the Bayesian evidence as the relative weighting. Doing so, we can compute the marginalized spectrum and KL divergence as shown in \cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps}. We find that the observable window is now clearly defined, and hints of the low-$k$ features survive this marginalization.
\subsection*{Historical context}
\Cref{fig:inflation_wh_utils/figures/pps_historical} shows reconstructions using the same methodology~\footnote{For the historical data prior to WMAP, we needed to significantly widen the priors on cosmological late-time parameters} but now on data from a historical sequence of CMB experiments
\begin{enumerate}
\item COBE~\cite{COBE_data},
\item ``pre-WMAP'' (COBE~\cite{COBE_data}, BOOMERANG~\cite{BOOMERANG_data}, MAXIMA~\cite{MAXIMA_data}, DASI~\cite{DASI_data}, VSA~\cite{VSA_data} and CBI~\cite{CBI_data}),
\item WMAP~\cite{bennett2012,hinshaw2012},
\item \textit{Planck}{} 2013 (TT+lowlike+lensing)~\cite{planck_2013} ,
\item \textit{Planck}{} 2015 (TT+lowTEB+lensing)~\cite{planck_2015} ,
\item \textit{Planck}{} 2018 (TT,TE,EE+lowE+lensing)~\cite{planck_legacy}.
\end{enumerate}
The hint of an $20<\ell<30$ feature becomes visible after WMAP, is strengthened in switching to \textit{Planck}{}, and remains stable as the \textit{Planck}{} data are updated.
Examining the KL divergences in \cref{fig:inflation_wh_utils/figures/pps_historical}, the vertical axis shows a greater overall constraint on the primordial power spectrum as improved cosmological constraints are obtained, and the horizontal axis shows that the $k$-window increases as the angular resolution of the experiments increases. The only alteration to this trend is the change from \textit{Planck}{} 2013 to \textit{Planck}{} 2015. In this case, the constraint on the primordial power spectrum is actually lowered, whilst the $k$-window increases. This is due to the fact that the $\tau$ constraint widened from 2013 to 2015, as can be seen it the top right panels of the \cref{fig:inflation_wh_utils/figures/pps_historical}.
\textit{Planck}{} 2018 provides the best constraints on the primordial power spectrum, both via its high-accuracy measurement of $\tau$, and in its small-scale angular resolution.
\FloatBarrier
\onecolumngrid
\begin{center}
\includegraphics{planck_figures_PPSR_TTTEEEv22_lowl_simall_b4_figures_pps_l_both_grid}
\captionof{figure}{Equally-weighted sample plots of primordial power spectrum reconstructions, conditioned on the number of knots $N$. The outermost knots are fixed at the bounds of the figure, so $N=1$ is equivalent to a scale-invariant primordial power spectrum, $N=2$ is equivalent to $\Lambda$CDM, up to a small difference in prior and $N>2$ has $N-2$ knots capable of moving in both the $k$ and $\mathcal{P}$ directions. Prior samples are drawn in red, whilst posterior samples are indicated in black.\label{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid}}
\end{center}
\begin{center}
\includegraphics{planck_figures_PPSR_TTTEEEv22_lowl_simall_b4_figures_pps_both_grid}
\captionof{figure}{Same as \cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid}, but plotted using iso-probability credibility intervals as discussed in \cref{sec:background_functional_inference}. Blue and red contours represent prior and posterior respectively.\label{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps_both_grid}}
\end{center}
\begin{center}
\includegraphics[width=0.495\textwidth]{planck_figures_PPSR_TTTEEEv22_lowl_simall_b4_figures_pps_l_both}
\includegraphics[width=0.495\textwidth]{planck_figures_PPSR_TTTEEEv22_lowl_simall_b4_figures_pps_both}
\includegraphics[width=0.495\textwidth]{planck_figures_PPSR_TTTEEEv22_lowl_simall_b4_figures_pps_evidence}
\includegraphics[width=0.495\textwidth]{planck_figures_PPSR_TTTEEEv22_lowl_simall_b4_figures_pps_dkl}
\captionof{figure}{{\em Bottom-left:\/} Bayesian evidence as a function of number of knots $N$ for the primordial power spectrum reconstruction. {\em Top:\/} Marginalized primordial power spectrum plot. These are produced by taking \cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid,fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps_both_grid} and weighting each panel by their respective evidence. {\em Bottom-right:\/} Marginalized conditional Kullback-Leibler divergence.
\label{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps}}
\end{center}
\begin{center}
\includegraphics{planck_figures_inflation_wh_utils_figures_pps_dkl_historical}
\includegraphics[width=0.49\textwidth]{planck_figures_tau}
\includegraphics{planck_figures_inflation_wh_utils_figures_pps_historical}
\captionof{figure}{Historical primordial power spectrum reconstructions. {\em Top-left:\/} Conditional Kullback-Leibler divergences. {\em Top-right:\/} inflationary power spectrum summary parameters, and the influence of $\tau$ on \textit{Planck}{} constraints. {\em Bottom:\/} Marginalized power spectrum plots for each dataset.\label{fig:inflation_wh_utils/figures/pps_historical}}
\end{center}
\clearpage
\newpage
\twocolumngrid
\section{Inflationary Potential reconstruction\label{sec:vphi_reconstruction}}
\FloatBarrier
\begin{figure}[t]
\includegraphics{spline}
\caption{For the inflationary potential reconstruction, we parameterize the second derivative of the logarithmic potential via a linear interpolating spline, and then integrate twice to recover the logarithmic potential (seen schematically from bottom to top of the figure). This introduces two additional parameters: a gradient and global offset.\label{fig:spline}}
\end{figure}
\begin{table}[t]
\begin{tabular}{lll}
Parameters& Prior type& Prior range \\
\hline
$N$ & Discrete Uniform & $[0, 8]$ \\
$\ln V_*$ & Uniform & $[-25, -15]$ \\
$\frac{\d{\ln V_*}}{\d{\phi}}$ & Log-Uniform & $[10^{-3}, 10^{-0.3}]$ \\
$\frac{\d[2]{\ln V_1}}{\d{\phi}^2},\ldots,\frac{\d[2]{\ln V_N}}{\d{\phi}^2}$ & Uniform & $[-0.5,0.5]$ \\
$\phi_1,\ldots,\phi_N$ & Sorted uniform & $[\tilde{\phi}_{\min{}},\tilde{\phi}_{\max{}}]$ \\
$\ln 10^{10}\mathcal{P}_\mathcal{R}(k)$ & Indirect constraint & $[2,4]$ \\
\end{tabular}
\caption{The prior distributions on early-time cosmological parameters for the inflationary potential reconstruction. $\tilde{\phi}_{\min{}}$ and $\tilde{\phi}_{\max{}}$ are defined by the observable window of the unperturbed potential. There is a further prior constraint in that we require that the inflaton should evolve in an inflating phase throughout the observable window and that the inflaton should be rolling downhill from negative to positive $\phi$ throughout.\label{tab:vphi_prior}}
\end{table}
In contrast to the analysis from the previous section, instead of parameterising the primordial power spectrum directly, here we take the pipeline one stage backward and perform a non-parametric reconstruction of the inflaton potential $V(\phi)$. The scalar and tensor primordial power spectra $\mathcal{P}_{\mathcal{R},\mathcal{T}}$ are then derived from $V(\phi)$ via the procedure indicated in \cref{sec:background_primordial_cosmology}.
To reconstruct the inflationary potential $V(\phi)$, it is more appropriate to work first with $\ln V$, as in general $V(\phi)$ can {\em a-priori\/} span a many scales, and it is typically $\frac{\d{}}{\d{\phi}}\ln V$ that drives much of the evolution of the inflaton during inflation.
More importantly, one cannot parameterize $V(\phi)$ via a linear interpolating spline as was the case in \cref{sec:PPSR}. The equations of motion (\cref{eqn:klein_gordon,eqn:Friedmann,eqn:mukhanov_sazaki_scalar,eqn:mukhanov_sazaki_tensor}) in general depend on first (and sometimes second) derivatives of $V(\phi)$. Parameterizing the potential using a linear spline will typically yield primordial power spectra with (arguably) unphysical ringing effects.
One should therefore use a spline with continuous first derivatives, and it is natural to choose a cubic spline as the conceptually simplest smooth interpolator. It is tempting to try to do this directly by taking the locations of the knots of the spline as free parameters. Cubic splines, however, are stiff, yielding complicated posteriors that are very difficult to navigate, interpret, and set priors on~\cite{Vazquez_primordial}.
Cubic splines have the property that their derivative is a smooth piecewise quadratic, and their second derivative is a piecewise linear spline. This suggests that the cleanest way to reconstruct the potential is to parameterize the second derivative as a linear spline, and then integrate this function twice to get the log-potential. Two additional parameters are created by this double integration, a gradient term $\frac{\d{\ln V_*}}{\d{\phi}}$ and an overall offset $\ln V_*$. These two free parameters function as an alternative constraint choice in comparison with natural or clamped splines. Our reconstruction function is therefore
\begin{align}
\ln V =& \ln V_* + (\phi-\phi_*)\frac{\d{\ln V_*}}{\d{\phi}}
\nonumber\\
&+ \int_{\phi_*}^\phi\d{{\phi^\prime}}\int_{\phi_*}^{\phi^\prime} \d{\phi^{\prime\prime}} \:\mathrm{Lin}(\phi^{\prime\prime};\Theta_V)\\
\Theta_V =& (\phi_1,\ldots,\phi_N,\frac{\d[2]{\ln V_1}}{\d{\phi}^2},\ldots,\frac{\d[2]{\ln V_N}}{\d{\phi}^2}, \frac{\d{\ln V_*}}{\d{\phi}}, \ln V_*),\nonumber
\end{align}
which can be viewed as working through \cref{fig:spline} in reverse.
\subsection*{Priors}
The priors in this analysis proved to be critically important for recovering sensible results. To harmonize with the analysis of the primordial power spectrum, our first requirement is that any primordial power spectrum generated from a potential $V(\phi)$ resides in the range ${2 < \ln 10^{10} \mathcal{P}_\mathcal{R}(k) < 4}$.
Consider the slow roll parameters~\cite{slow_roll}, and their relation to the second derivative of the log potential
\begin{equation}
\varepsilon_V = \frac{1}{2}{\left(\frac{1}{V}\frac{\d{V_*}}{\d{\phi}}\right)}^2,\quad
\eta_V = \frac{1}{V}\frac{\d[2]{V_i}}{\d{\phi}^2}
\Rightarrow \frac{\d[2]{\ln V}}{\d{\phi}^2} = \eta_V - 2\varepsilon_V.
\nonumber
\end{equation}
We therefore take the priors on the second derivatives of the log potential $\frac{\d[2]{\ln V_i}}{\d{\phi}^2}$ to be uniformly distributed, and the gradient $\frac{\d{\ln V_*}}{\d{\phi}}$ is taken to be negatively log-uniform. Negativity forces the inflaton to roll downhill from negative to positive $\phi$, breaking a symmetric degeneracy. We take the potential offset to vary across a wide range $\ln V_*$. Widening any of these priors detailed in \cref{tab:vphi_prior} further has no effect, as any primordial power spectrum generated outside these bounds lies outside the range $[2,4]$.
Particular care must be taken with the horizontal locations of the knots. Any reconstruction of the potential will be sensitive only to the observable window of inflation $[\phi_{\min{}},\phi_{\max{}}]$, defined as when the largest and smallest observable scales $k_{\min{}}$ and $k_{\max{}}$ exit the horizon.
As in \cref{sec:PPSR}, we take $(k_{\min{}},k_{\max{}})=(10^{-4},10^{-0.3})\mathrm{Mpc}^{-1}$.
Unfortunately, the bounds of the window $[\phi_{\min{}},\phi_{\max{}}]$ are strongly dependent on the other primordial parameters. One cannot therefore take an arbitrarily wide range in $\phi$ for the horizontal locations, as the reconstruction is then dominated by the prior effect of unconstrained knot parameters.
The locations $\phi_1,\ldots,\phi_N$ of the reconstruction knots should instead be distributed throughout the observable window. Whilst the locations $\phi_1,\ldots\phi_N$ and heights $\frac{\d[2]{\ln V_1}}{\d{\phi}^2},\ldots,\frac{\d[2]{\ln V_N}}{\d{\phi}^2}$ themselves influence the size of the observable window, a reasonable approach is to first estimate it using the unperturbed potential (i.e.\ setting $N=0$), giving an alternative window $[\tilde{\phi}_{\min{}},\tilde{\phi}_{\max{}}]$. In a similar manner as the horizontal knots in \cref{sec:PPSR}, we take the $N$ horizontal $\phi$-knot locations to be sorted and uniform throughout this window.
Finally, we require that the inflaton should evolve in an inflating phase throughout the observable window, and that it should be rolling (not necessarily in slow roll) downhill from negative to positive $\phi$ throughout. These priors are summarized in \cref{tab:vphi_prior}.
Alternative methodologies for direct reconstruction of the potential exist in the literature. One approach is to to expand the potential $V(\phi)$ as a Taylor series~\citep{Lesgourgues_potential,Hiranya_potential_1}. Another is to expand $H(\phi)$ as a Taylor series~\citep{Lesgourgues_potential_2,Hiranya_potential_3}, and then derive the potential analytically via ${V(\phi) = 3M_\p^2H^2-2M_\p^4{H^\prime}^2}$. Both of these approaches have been successfully applied in the \textit{Planck}{} inflation papers~\cite{planck_inflation_2013,planck_inflation_2015,planck_inflation}.
\vspace{-1.5em}
\subsection*{Results}
Due to the strong dependency of the $\phi$-window on the potential itself, it is not particularly illuminating to plot $V(\phi)$ directly. Instead, in the spirit of the other two sections we start by plotting the functional posterior of the primordial power spectrum $\mathcal{P}_\mathcal{R}(k)$, shown in \cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid,fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps}. Viewed in this manner, one can think of these primordial power spectrum reconstructions as having an alternative prior complementary to \cref{sec:PPSR}, motivated by the assumption that the primordial power spectrum is derived from a smooth underlying potential.
In the same manner as \cref{sec:PPSR}, \cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid} and is consequently a form of exponential potential. Regardless, it recovers a primordial power spectrum with an appropriate amplitude and tilt and minimal running, almost identical to the traditional $A_\mathrm{s},n_\mathrm{s}$ parameterization. $N=1$ adds a constant second derivative term to the Taylor expansion, and produces a similar primordial power spectrum. As more knots are added, the potential has greater freedom, and the corresponding primordial power spectrum begins to gain similar features to the results in \cref{sec:PPSR}; a loss of constraint at low and high-$k$. Intriguingly, there is also the same preference for an oscillation with a peak at $\ell\sim50$ and trough at $20<\ell<30$. In the line plots of \cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid} the oscillation is now smooth, on account of the physical potential-based prior created by this reconstruction. It should be noted that while such oscillations are characteristic of this integrated inflationary potential parameterization, these were also partially recovered {\em a priori\/} in the free-form approach from \cref{sec:PPSR}.
Examining the evidences in \cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps}, we can see that despite the similarities in primordial power spectra $N=1$ is preferred over $N=0$. The reason for this is that the restrictive form of potential for $N=0$ forces $r\approx0.2$, which is now ruled out by \textit{Planck}{}. Allowing a second derivative for the $N=1$ relaxes the $r$ constraint, resulting in a Bayesian preference for the $N=1$ case, consistent with the results of the~\citet{planck_inflation}. Adding further knots causes the evidence to drop, indicating that from a Bayesian standpoint, no further complexity is required by the data. The marginalized plots in \cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps} show similar attributes to those of the primordial power spectrum in \cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps}, but in this case the stiffness of the primordial power spectrum reconstruction results in a slightly poorer recovery of the relative lack of power spectrum constraint at low and high-$k$.
Our second set of plots detail results for the inflationary slow roll parameter $\eta_V(k)$, shown in \cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/etaV_l_both_grid,fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/etaV}. Instead of using $\phi$ as the independent variable (which suffers from the dependency of the window widths on the underlying potential), we use the effective wavenumber $k(\phi)$, which is monotonically related to $\phi$ in our reconstruction, defined to be the size of the comoving Hubble radius at that moment in the field's evolution.
\Cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/etaV_l_both_grid} reveals that the oscillations in the primordial power spectrum at low $k$ are created by a partial breakdown in the slow roll conditions. For $N\ge5$, $\eta_V\sim 0.5$, which are the same values of $N$ at which oscillations become apparent in \cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid}. This will be of particular interest for just enough inflation models~\citep{just_enough_inflation,kinetic_dominance,Hergt1,Hergt2}, models with singularities and discontinuities~\cite{H1,H2,H3}, multi-field phase-transitions~\cite{H4,H5,H6,H7,H8}, M-theory~\cite{H9,H10} or supergravity~\cite{H11} models, to name a few examples. There is a long history of confronting such models with data~\cite{WMAP3, H12, H13, H14, H15, H16, H17, H18}. \Cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/etaV} details the marginalized results.
\textit{Planck}{} provides only a weak upper bound on the other slow roll parameter $\varepsilon_V\approx \frac{r}{16}$, meaning $\varepsilon_V$ is nearly indistinguishable from its logarithmic prior. Given the slow roll relations, we find that for our reconstructions $\eta_V\approx\frac{\d[2]{\log V}}{\d{\phi}^2}$, so plots of $\eta_V$ are nearly identical to plots of the equivalent underlying linear second-derivative reconstruction parameters.
\FloatBarrier
\onecolumngrid
\begin{center}
\includegraphics[width=0.495\textwidth]{planck_figures_vphi_reconstruction_TTTEEEv22_lowl_simall_b4_figures_pps_l_both}
\includegraphics[width=0.495\textwidth]{planck_figures_vphi_reconstruction_TTTEEEv22_lowl_simall_b4_figures_pps_both}
\includegraphics[width=0.495\textwidth]{planck_figures_vphi_reconstruction_TTTEEEv22_lowl_simall_b4_figures_pps_evidence}
\includegraphics[width=0.495\textwidth]{planck_figures_vphi_reconstruction_TTTEEEv22_lowl_simall_b4_figures_pps_dkl}
\captionof{figure}{{\em Bottom-left:\/} Bayesian evidence as a function of number of knots $N$ for the inflationary potential reconstruction. {\em Top:\/} Marginalized functional posteriors for the primordial power spectrum. These are produced by taking \cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid} and weighting each panel by their respective evidence. {\em Bottom-right:\/} Marginalized conditional Kullback-Leibler divergence.\label{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps}}
\end{center}
\begin{center}
\includegraphics{planck_figures_vphi_reconstruction_TTTEEEv22_lowl_simall_b4_figures_pps_grid_both}
\vspace{-2em}
\captionof{figure}{Equally-weighted sample plots of the functional posterior of the primordial power spectrum from the inflationary potential reconstruction, conditioned on the number of knots $N$. $N=0,1$ have a potential equivalent to a first and second-order Taylor expansion respectively, whilst $N\ge 2$ provide the ability to reconstruct broad features in the underlying potential. Prior samples are drawn in red, whilst posterior samples are indicated in black.\label{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid}}
\end{center}
\begin{center}
\includegraphics[width=0.495\textwidth]{planck_figures_vphi_reconstruction_TTTEEEv22_lowl_simall_b4_figures_etaV_l_both}
\includegraphics[width=0.495\textwidth]{planck_figures_vphi_reconstruction_TTTEEEv22_lowl_simall_b4_figures_etaV_both}
\includegraphics[width=0.495\textwidth]{planck_figures_vphi_reconstruction_TTTEEEv22_lowl_simall_b4_figures_etaV_evidence}
\includegraphics[width=0.495\textwidth]{planck_figures_vphi_reconstruction_TTTEEEv22_lowl_simall_b4_figures_etaV_dkl}
\captionof{figure}{{\em Bottom-left:\/} Bayesian evidence as a function of number of knots $N$ for the inflationary potential reconstruction. {\em Top:\/} Marginalized functional posterior of the inflationary parameter $\eta_V$. These are produced by taking \cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/etaV_l_both_grid} and weighting each panel by their respective evidence. {\em Bottom-right:\/} Marginalized conditional Kullback-Leibler divergence.\label{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/etaV}}
\end{center}
\begin{center}
\includegraphics{planck_figures_vphi_reconstruction_TTTEEEv22_lowl_simall_b4_figures_etaV_grid_both}
\captionof{figure}{Same as \cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid}, but now for the inflationary slow roll parameter $\eta_V(k)$, with independent variable defined by an effective wavenumber $k$, which sits in one-to-one correspondence with $\phi$ via the size of the comoving Hubble radius at that moment in the field's evolution.\label{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/etaV_l_both_grid}}
\end{center}
\newpage
\clearpage
\twocolumngrid
\section{Sharp feature reconstruction\label{sec:features}}
\FloatBarrier
In this section, we return to a direct analysis of the primordial power spectrum. Inspired by the oscillatory features present in both the linear spline primordial power spectrum reconstruction (\cref{sec:PPSR}) and in the functional posterior of the primordial power spectrum from the inflationary potential reconstruction (\cref{sec:vphi_reconstruction}), we now consider a parameterization of the primordial power spectrum which favors sharp features. Attempts at explaining these features has a long history in the literature, initially investigated in Refs.~\cite{WMAP1,WMAP2,WMAP3,Features}.
We introduce sharp features into the parameterization of the spectrum by placing a variable number $N$ of top-hat functions with varying widths $\Delta$, heights $h$, and locations $\log_{10}k$ on top of the traditional \(A_\mathrm{s}, n_\mathrm{s}\) parameterization
\begin{align}
\ln \mathcal{P}_\mathcal{R}(k) =& \ln A_\mathrm{s} + (n_\mathrm{s} -1) \ln\left( \frac{k}{k_*} \right)\nonumber\\
&+ \sum\limits_{i=1}^N h_i \left[ |\log_{10}k-\log_{10}k_i|< \frac{\Delta_i}{2} \right],
\label{eqn:features_parametrisation}\\
\Theta_\mathcal{P} =& (\Delta_1,\cdots,\Delta_N,h_1,\cdots,h_N,k_1,\cdots,k_N). \nonumber
\end{align}
where the square brackets (as in \cref{sec:background_non_parametric}) denote a logical truth function~\citep{ConcreteMathematics}. This parameterisation is indicated schematically in \cref{fig:pps_features_parametrisation}, and \cref{tab:features_prior} provides a summary of the priors that we use. Readers are referred to the priors sections of \cref{sec:PPSR,sec:vphi_reconstruction} for further details.
\subsection*{Results}
The results for the sharp feature reconstructions can be found in \cref{fig:features_log/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid,fig:features_log/TTTEEEv22_lowl_simall_b4/figures/pps,fig:features_log/TTTEEEv22_lowl_simall_b4/figures/cl}. \Cref{fig:features_log/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid} shows that in the marginalized plots the oscillations and lack of power spectrum constraints at low- and high-$k$ are once again recovered. Visually, the features are even more striking in these reconstructions, on account of the ability for this parameterization to localize in $k$ more precisely. There are also hints of features at high-$k$ in this reconstruction, which were smoothed out by the parameterizations of the two previous sections.
Marginalization in \cref{fig:features_log/TTTEEEv22_lowl_simall_b4/figures/pps} shows that there is little Bayesian evidence to support the introduction of more than two features, but the low-$k$ oscillation still comes through clearly in the fully marginalized plot.
Finally, we examine the effects of these reconstructions on the $C_\ell$ spectra by considering the functional posterior in \cref{fig:features_log/TTTEEEv22_lowl_simall_b4/figures/cl}. By comparing the $\Lambda$CDM case $N=0$ with the case $N=8$ we see that the features at low-$k$ in the PPS correspond to both a suppression of power at low-$\ell$ in the TT spectrum, as well as a more specific localized reduction of power in the $20<\ell<30$ region. There seems to be no obvious correspondence with possible features seen in the polarization TE or EE spectra.
\begin{figure}[ht]
\includegraphics{features}
\caption{For the sharp features reconstruction, we parameterize the primordial power spectrum via traditional amplitude-tilt $(A_\mathrm{s},n_\mathrm{s})$ parameterization, with $N$ top-hat features. We constrain the spectrum to be within the dashed box.\label{fig:pps_features_parametrisation}}
\end{figure}
\begin{table}[ht]
\begin{tabular}{lll}
Parameters& Prior type& Prior range \\
\hline
$N$ & discrete uniform & $[0, 8]$ \\
$A_s$ & uniform & $10^{-10}[e^2,e^4]$ \\
$n_s$ & uniform & $[0.8,1.2]$ \\
$h_1,\cdots,h_N$ & uniform & $[-1,1]$\\
$k_2<\cdots<k_{N-1}$ & sorted $\log$-uniform & $[10^{-4},10^{-0.3}]$ \\
$\Delta_1,\cdots,\Delta_N$ & uniform & $ [0,1] $\\
$\ln 10^{10}\mathcal{P}_\mathcal{R}(k)$ & Indirect constraint & $[2,4]$ \\
\end{tabular}
\caption{The prior distributions on early-time cosmological parameters for the sharp feature reconstructions.\label{tab:features_prior}}
\end{table}
\FloatBarrier
\onecolumngrid
\begin{center}
\includegraphics[width=0.495\textwidth]{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_pps_l_both}
\includegraphics[width=0.495\textwidth]{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_pps_both}
\includegraphics[width=0.495\textwidth]{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_pps_evidence}
\includegraphics[width=0.495\textwidth]{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_pps_dkl}
\captionof{figure}{{\em Bottom-left:\/} Bayesian evidence as a function of number of knots $N$ for the sharp features reconstruction. {\em Top:\/} Marginalized primordial power spectrum plot. These are produced by taking \cref{fig:features_log/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid} and weighting each panel by their respective evidence. {\em Bottom-right:\/} Marginalized conditional Kullback-Leibler divergence.
\label{fig:features_log/TTTEEEv22_lowl_simall_b4/figures/pps}}
\end{center}
\begin{center}
\includegraphics{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_pps_grid_both}
\captionof{figure}{Equally-weighted sample plots of sharp features reconstruction, conditioned on the number of knots $N$. $N=0$ is exactly equivalent to a standard $\Lambda$CDM parameterization. Prior samples are drawn in red, whilst posterior samples are indicated in black.\label{fig:features_log/TTTEEEv22_lowl_simall_b4/figures/pps_l_both_grid}}
\end{center}
\begin{center}
\includegraphics[width=0.495\textwidth]{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_clTT_0}
\includegraphics[width=0.495\textwidth]{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_clTT_8}
\includegraphics[width=0.495\textwidth]{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_clTE_0}
\includegraphics[width=0.495\textwidth]{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_clTE_8}
\includegraphics[width=0.495\textwidth]{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_clEE_0}
\includegraphics[width=0.495\textwidth]{planck_figures_features_log_TTTEEEv22_lowl_simall_b4_figures_clEE_8}
\captionof{figure}{Functional posterior distribution for the $C_\ell$ spectra from the sharp features reconstruction. Left hand column of panels is for $N=0$, i.e.\ a $\Lambda$CDM parameterization. Right hand column of panels is for $N=8$ features. Residual plots are with respect to the \textit{Planck}{} 2018 best-fit $\Lambda$CDM cosmology.
\label{fig:features_log/TTTEEEv22_lowl_simall_b4/figures/cl}}
\end{center}
\begin{center}
\includegraphics{planck_figures_parameters}
\captionof{figure}{Stability of the cosmological parameters for the primordial power spectrum reconstruction (PPS), the potential reconstruction (V) and the sharp features reconstruction (SF). Parameters are shown for the reconstructions conditioned on $N$, and for the marginalized case. For all cases except the highly-disfavored $N=1$ PPS reconstruction (equivalent to an $n_s=1$ scale-invariant power spectrum), the parameters agree with the baseline $\Lambda$CDM parameters. Note that for PPS and V the parameters $n_s$ and $A_s$ are derived parameters.\label{fig:planck_figures/parameters}}
\end{center}
\clearpage
\newpage
\twocolumngrid
\section{Cosmological parameter stability}
Finally in \cref{fig:planck_figures/parameters} we show that the underlying cosmological parameter constraints remain effectively unchanged for all three of the analyses in \cref{sec:PPSR,sec:vphi_reconstruction,sec:features}, in spite of the additional degrees of freedom we have given to the primordial power spectrum.
The only exception is the PPS $N=1$ reconstruction. This model is highly-disfavored (\cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps}), since \textit{Planck}{} rules out a Harrison-Zeldovich scale-invariant ($n_s=1$) spectrum~\cite{planck_inflation_2013,planck_2015,planck_inflation}. Requiring $n_s=1$ gives a poorly-fit model which forces the cosmological parameters into locations discordant with $\Lambda$CDM.
This overall parameter stability for models consistent with the data demonstrates that one can explain features in CMB power spectra via modifications to the primordial cosmology, without the need to alter late-time cosmological parameters.
\section{Conclusions\label{sec:conclusion}}
In this work, we have reconstructed the primordial Universe three ways. In \cref{sec:PPSR}, we reconstructed the primordial power spectrum using a linear spline. In \cref{sec:vphi_reconstruction}, we reconstructed the inflationary potential using a cubic spline. In \cref{sec:features}, we probed sharp features in the primordial power spectrum by superimposing top-hat functions on top of the traditional $\Lambda$CDM power spectrum.
We showed that the Bayesian odds of a scale-invariant power spectrum are around a quintillion to one against in comparison to the $\Lambda$CDM cosmology. This agrees with the Planck Collaboration's conclusions~\cite{planck_inflation} that there is decisive evidence for $n_\mathrm{s}\ne 1$ --- one of the key predictions of the theory of inflation.
All methods reconstruct a featureless tilted power law consistent with a simple $(A_\mathrm{s},n_\mathrm{s})$ parameterization across a broad observable window $(50\lesssim\ell\lesssim2000)$. In addition, all reconstructions demonstrate that in a Bayesian sense it is preferable to have models which are able to recover the lack of power spectrum constraints at low-$k$ due to cosmic variance, and at high-$k$ due to \textit{Planck}{} instrument noise, reflected in the evidences and marginalized plots (\cref{fig:PPSR/TTTEEEv22_lowl_simall_b4/figures/pps,fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/pps,fig:features_log/TTTEEEv22_lowl_simall_b4/figures/pps}).
All large $N$ conditional reconstructions partially recover oscillatory features in the primordial power spectrum, with a peak at $\ell\sim50$ and a trough at $20<\ell<30$, which manifest themselves in the functional posteriors of the $C_\ell$ spectra (\cref{fig:features_log/TTTEEEv22_lowl_simall_b4/figures/cl}). The inflationary potential reconstruction (\cref{sec:vphi_reconstruction}) shows that this oscillation could be due to a breakdown in slow roll near the start of the inflationary window (\cref{fig:vphi_reconstruction/TTTEEEv22_lowl_simall_b4/figures/etaV}), which is relevant for a wide variety of inflationary models~\citep{just_enough_inflation,kinetic_dominance,Hergt1,Hergt2,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,WMAP3,H12,H13,H14,H15,H16,H17,H18}. However, the oscillations do not survive marginalization over $N$, indicating that the Bayesian evidence is not strong enough from \textit{Planck}{} data to indicate a significant detection of such a feature.
The renewed upper bound on $r$ from \textit{Planck}{} 2018 now has enough discriminatory power to begin reconstructing potentials, as shown by the preference for the $N=1$ case in the inflationary potential reconstructions.
As shown in \cref{fig:planck_figures/parameters}, in all cases, the distributions on the late-time cosmological parameters remain unperturbed by the additional degrees of freedom on the primordial cosmology provided by these reconstructions, indicating that any conclusions using late-time parameters are unlikely to be affected by modifying the primordial cosmology.
There is scope for inflationary models which {\em a-priori\/} predict these low-$k$ features to be preferred over the $\Lambda$CDM cosmology, particularly if such models are capable of producing sharper features in the $C_\ell$ spectra at $20<\ell<30$. Additionally, in light of further CMB data~\cite{Core_inflation}, or failing that, strong $\tau$ characterization, it is likely that these hints of features will sharpen and provide further discriminatory power in constructing better models of the primordial Universe.
\begin{acknowledgments}
WJH was supported via STFC PhD studentship RG68795, the European Research Council (ERC) under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement number 306478-CosmicDawn, and a Gonville \& Caius College Research Fellowship.
WJH and ANL thank Fabio Finelli, Martin Bucher and Julien Lesgourgues for numerous useful comments and suggestions for improvements over the course of this work.
HVP acknowledges useful conversations with George Efstathiou, Matt Johnson and Keir Rogers. HVP was supported by the European Research Council (ERC) under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement number 306478-CosmicDawn. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation Grant PHY-1607611. This work was also partially supported by a grant from the Simons Foundation.
This work was performed using the Darwin Supercomputer of the \href{http://www.hpc.cam.ac.uk/}{University of Cambridge High Performance Computing Service}, provided by Dell Inc.\ using Strategic Research Infrastructure Funding from the Higher Education Funding Council for England and funding from the Science and Technology Facilities Council, as well as resources provided by the \href{http://www.csd3.cam.ac.uk/}{Cambridge Service for Data Driven Discovery (CSD3)} operated by the University of Cambridge Research Computing Service, provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/P020259/1), and \href{www.dirac.ac.uk}{DiRAC funding from the Science and Technology Facilities Council}.
\end{acknowledgments}
\vfill
\pagebreak
\bibliographystyle{unsrtnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,128 |
{"url":"http:\/\/www.gamedev.net\/index.php?app=forums&module=extras§ion=postHistory&pid=5129507","text":"\u2022 Create Account\n\n### #Actualfir\n\nPosted 07 February 2014 - 01:10 AM\n\nAs everyone suspected, it's just a Moir\u00e9 pattern. Here's my attempt at producing it:\n\n#include <cstdio>\n#include <cmath>\n\nint main() {\nstd::puts(\"# ImageMagick pixel enumeration: 800,800,255,srgb\");\n\nfor (int j=0; j<800; ++j) {\ndouble y = (400.0 - j) \/ 360.0;\nfor (int i=0; i<800; ++i) {\ndouble x = (i - 400.0) \/ 360.0;\ndouble z2 = 1.0 - x * x - y * y;\ndouble distance = (z2 >= 0.0) ? std::sqrt(z2) : 0.0;\ndouble color = std::fmod(1000.0*distance, 1.0);\nint r = 256 * color;\nint g = 256 * color;\nint b = 256 * (1.0 - color);\nr = r > 255 ? 255 : r < 0 ? 0 : r;\ng = g > 255 ? 255 : g < 0 ? 0 : g;\nb = b > 255 ? 255 : b < 0 ? 0 : b;\nstd::printf(\"%d,%d: (%d,%d,%d)\u00a0 #%02X%02X%02X\u00a0 srgb(%d,%d,%d)\\n\",\ni, j,\nr, g, b,\nr, g, b,\nr, g, b);\n}\n}\n}\n\n\n\nI compiled that code and then executed it, passing the output through | convert TXT:- output.png' (convert' is a command-line utility, part of ImageMagick). The output is this:\n\n[EDIT: If you replace 1000.0 with something like 250.0, you'll get an image much closer to the original in this thread.]\n\nVery good work! I see the central part is probably\n\ndistance = sqrt(1-(x*x+y*y))\n\ndo you know maybe what klind of function it is id drawed z=f(x,y)\n\nof just z=f(x,0); ?\n\nIm rarely doing mathematics so i forgot the thing\n\nAs to moire pattern I suspect this could be treated set of infinite number of moire interferentions - but those interferentions are purely mathematical not 'presentation aliasing' artifacts\n\n### #1fir\n\nPosted 07 February 2014 - 01:06 AM\n\nAs everyone suspected, it's just a Moir\u00e9 pattern. Here's my attempt at producing it:\n\n#include <cstdio>\n#include <cmath>\n\nint main() {\nstd::puts(\"# ImageMagick pixel enumeration: 800,800,255,srgb\");\n\nfor (int j=0; j<800; ++j) {\ndouble y = (400.0 - j) \/ 360.0;\nfor (int i=0; i<800; ++i) {\ndouble x = (i - 400.0) \/ 360.0;\ndouble z2 = 1.0 - x * x - y * y;\ndouble distance = (z2 >= 0.0) ? std::sqrt(z2) : 0.0;\ndouble color = std::fmod(1000.0*distance, 1.0);\nint r = 256 * color;\nint g = 256 * color;\nint b = 256 * (1.0 - color);\nr = r > 255 ? 255 : r < 0 ? 0 : r;\ng = g > 255 ? 255 : g < 0 ? 0 : g;\nb = b > 255 ? 255 : b < 0 ? 0 : b;\nstd::printf(\"%d,%d: (%d,%d,%d)\u00a0 #%02X%02X%02X\u00a0 srgb(%d,%d,%d)\\n\",\ni, j,\nr, g, b,\nr, g, b,\nr, g, b);\n}\n}\n}\n\n\n\nI compiled that code and then executed it, passing the output through | convert TXT:- output.png' (convert' is a command-line utility, part of ImageMagick). The output is this:\n\n[EDIT: If you replace 1000.0 with something like 250.0, you'll get an image much closer to the original in this thread.]\n\nVery good work! I see the central part is probably\n\ndistance = sqrt(1-(x*x+y*y))\n\ndo you know maybe what klind of function it is id drawed z=f(x,y)\n\nof just z=f(x,0); ?\n\nIm rarely doing mathematics so i forgot the thing\n\nPARTNERS","date":"2014-08-27 09:06:11","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\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3912941515445709, \"perplexity\": 7377.535553429664}, \"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-2014-35\/segments\/1408500828050.28\/warc\/CC-MAIN-20140820021348-00373-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
module.exports = require('./lib/patternEmitter');
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,760 |
Q: Do continuous functions with small uniform norm preserve order? Consider a sequence $x_1,\dots,x_n\in [0,1]^d$. Suppose that $\mathcal{F}$ is a subset of the space of continuous functions from $[0,1]^d$ to $\mathbb{R}$.
Moreover, let $f\in\mathcal{F}$ such that
$$f(x_1)< f(x_2)< \dots < f(x_n).$$
I'm wondering if there is any condition over $\mathcal{F}$ and $\delta>0$ so we can conclude that for all $g\in\mathcal{F}$ with $\|f-g\|_{\infty}<\delta$,
$$g(x_1)< g(x_2)< \dots < g(x_n).$$
I think this may be true if we take $f,g$ lipschitz and something like $\delta=\min_{i,j}\|x_i-x_j\|$, but I'm not sure.
Thank you very much.
Edit: I relaxed changed the condition
$$f(x_1)\leq f(x_2)\leq \dots \leq f(x_n),$$
to
$$f(x_1)< f(x_2)< \dots < f(x_n).$$
A: With strictly increasing values $f(x_1) < f(x_2) < \cdots < f(x_n)$ you can define $\epsilon = \frac 12 \min(f(x_{k+1})-f(x_k))$. Then $\Vert f-g \Vert_\infty < \epsilon$ implies
$$
g(x_k) < f(x_k) + \epsilon \le f(x_{k+1}) - \epsilon < g(x_{k+1})
$$
for $1 \le k < n$.
This works for all bounded functions $f, g$ from an arbitrary domain to $\Bbb R$, the continuity is not relevant here.
Requiring Lipschitz continuity for $f$ does not help because that does not guarantee a minimum distance between the $f(x_k)$.
A: Let's consider the case $n=2$ that can be easily generalized.
Under the stronger condition $f(x_1) \lt f(x_2)$ the result holds.
For the proof, consider the map
$$\begin{array}{l|rcl}
\psi : & (\mathcal F, \Vert \cdot \Vert_\infty) & \longrightarrow & \mathbb R \\
& f & \longmapsto & f(x_2) - f(x_1) \end{array}$$
$\psi$ is linear, continuous as for $f, g \in \mathcal F$
$$\vert \psi(f) \vert \le 2 \Vert f \Vert_\infty.$$
Therefore if $\psi(f) = a \gt 0$,
$$\{g \in \mathcal F \mid \psi(g) \gt a/2\}$$ is open being the inverse image under $\psi$ of the open subset $(a/2,\infty)$. This allows to get the desired conslusion.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,423 |
Q: geom_smooth() with facet_grid() versus mgcv:::gam with by="" I am comparing the output from a gam performed through ggplot2's geom_smooth using facet_grid versus a gam using the mgcv:::gam with specified "by" factor visualized through visreg. Herewith data and code:
library(dplyr)
set.seed(1)
dat <- iris %>% mutate(response = sample(rep(c(0,1),length.out=150/2),150, replace=T))
#Just the output from geom_smooth
library(ggplot2)
ggplot(dat, aes(Sepal.Length,response)) +
geom_point() +
geom_smooth(method="gam", formula = y~s(x, bs="cs"), method.args=list("binomial")) +
facet_grid(.~Species)
#Now performing the gam through mgcv:::gam specifying by=Species
library(mgcv)
gam <- gam(dat, formula = response~s(Sepal.Length, bs="cs", by=Species),family = binomial())
#Comparing the two different outputs
library(visreg)
visreg(gam, "Sepal.Length", by="Species", scale="response", gg=T) +
guides(color=F)+
geom_smooth(data=dat,aes(Sepal.Length,response),
method="gam", formula = y~s(x, bs="cs"), method.args=list("binomial"), color="red", fill="green")
Basically, what I think happens is that the mgcv:::gam bases its gam smooths on some type of "imputed" data for areas where there is actually none for each Species level. It seems the setting in geom_smooth() avoids this. Does anyone know how to counteract this, so that the output from geom_smooth and mgcv:::gam is identical?
EDIT:
Based on the answer from user20650, the code was updated to:
library(mgcv)
gam <- gam(dat, formula = response~Species + s(Sepal.Length, bs="cs", by=Species),family = binomial())
library(visreg)
visreg(gam, "Sepal.Length", by="Species", scale="response", gg=T) +
guides(color=F)+
geom_smooth(data=dat,aes(Sepal.Length,response),
method="gam", formula = y~s(x, bs="cs"), method.args=list("binomial"), color="red", fill="green",
fullrange=T)
As can be seen in the above plot there are subtle differences (mostly in the CI) in the two methods. This is accentuated should we look at e.g. set.seed(100):
library(dplyr)
set.seed(100)
dat <- iris %>% mutate(response = sample(rep(c(0,1),length.out=150/2),150, replace=T))
library(mgcv)
gam <- gam(dat, formula = response~Species + s(Sepal.Length, bs="cs", by=Species),family = binomial())
library(visreg)
visreg(gam, "Sepal.Length", by="Species", scale="response", gg=T) +
guides(color=F)+
geom_smooth(data=dat,aes(Sepal.Length,response),
method="gam", formula = y~s(x, bs="cs"), method.args=list("binomial"), color="red", fill="green",
fullrange=T)
Can anyone explain the differences in the two methods, and how to generate identical output from geom_smooth() in mgcv:::gam and vice versa (both in the case of fullrange=T as well as fullrange=F)?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,382 |
body {
padding-top: 70px;
background-color: #eee;
}
.list-group-item {
height: 55px;
}
.form-signin {
max-width: 330px;
padding: 15px;
margin: 0 auto;
}
.form-signin .form-signin-heading,
.form-signin .checkbox {
margin-bottom: 10px;
}
.form-signin .checkbox {
font-weight: normal;
}
.form-signin .form-control {
position: relative;
height: auto;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
padding: 10px;
font-size: 16px;
}
.form-signin .form-control:focus {
z-index: 2;
}
.form-signin input[type="email"] {
margin-bottom: -1px;
border-bottom-right-radius: 0;
border-bottom-left-radius: 0;
}
.form-signin input[type="password"] {
margin-bottom: 10px;
border-top-left-radius: 0;
border-top-right-radius: 0;
}
.pointer {
cursor: pointer;
}
.nav, .pagination, .carousel, .panel-title a {
cursor: pointer;
}
.task-header {
font-size: 18px;
}
.task-title {
font-size: 18px;
vertical-align: middle;
line-height: 35px;
padding-bottom: 5px;
}
.task-buttons {
float: right;
}
.task-description ul {
padding-left: 15px;
list-style: disc;
}
.task-user {
font-size: 12px;
text-align: right;
font-style: italic;
}
.list-group-item {
min-height: 55px;
height: auto;
}
.panel-heading {
font-size: 18px;
vertical-align: middle;
line-height: 35px;
padding-bottom: 5px;
}
.page-header {
margin-top: 0;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,638 |
Q: Cakephp Input Select Multiple Selected problems with selected Im having some basic problems with that
echo $this->Form->select('Already.alredyselected',$listnom,array(
'multiple' => 'checkbox',
'label' => 'Text Label',
'selected' => $alreadyinvites,
'class' => 'selbox'));
My arrays are:
$alreadyinvites = Array(
[258] => Banana Voadora
[196] => Jack Bob Rodrigo Silva Junior)
and:
$listnom = Array(
[258] => Banana Voadora
[196] => Jack Bob Rodrigo Silva Junior
[259] => Toucinho Defumado
)
but the CakePHP 2.4 or 2.3.2 arent printing the Selected on checkbox, What im doing wrong ?
A: try
$alreadyinvites = array(258, 196);
and
echo $this->Form->select('Already.alredyselected',$listnom,array(
'multiple' => 'checkbox',
'label' => 'Text Label',
'value' => $alreadyinvites,
'class' => 'selbox'));
A: Try this:
$selected = array(2, 3);
$options = array(1, 2, 3, 4);
echo $this->Form->input('Attendees', array('multiple' => true, 'options' => $options, 'selected' => $selected));
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,664 |
Q: How to use the carousel component from Bootstrap 4 with React.js? Here is the following code that I was trying to get to work with the carousel => That's basically the example given on the site.
No Error, The active slider is shown but the sliding effect can't seem to work, I don't know how to make it work with React. Am I doing it wrong? Is it related to 'href'? Please Help me. It's driving me nuts...
import React from 'react';
export default class Slider extends React.Component {
render() {
return (
<div id="carouselExampleIndicators" className="carousel slide" data-ride="carousel">
<ol className="carousel-indicators">
<li data-target="#carouselExampleIndicators" data-slide-to="0" className="active"></li>
<li data-target="#carouselExampleIndicators" data-slide-to="1"></li>
<li data-target="#carouselExampleIndicators" data-slide-to="2"></li>
</ol>
<div className="carousel-inner" role="listbox">
<div className="carousel-item active">
<img className="d-block img-fluid" src="resources/img/cd-background-img.jpg" alt="First slide"/>
</div>
<div className="carousel-item">
<img className="d-block img-fluid" src="resources/img/home-bg.jpg" alt="Second slide"/>
</div>
<div className="carousel-item">
<img className="d-block img-fluid" src="resources/img/home-bg.jpg" alt="Third slide"/>
</div>
</div>
<a className="carousel-control-prev" href="#carouselExampleIndicators" role="button" data-slide="prev">
<span className="carousel-control-prev-icon" aria-hidden="true"></span>
<span className="sr-only">Previous</span>
</a>
<a className="carousel-control-next" href="#carouselExampleIndicators" role="button" data-slide="next">
<span className="carousel-control-next-icon" aria-hidden="true"></span>
<span className="sr-only">Next</span>
</a>
</div>
);
}
}
A: Try https://github.com/leandrowd/react-responsive-carousel. I am using it and works very well!
A: Here is another option, this is a full bootstrap components migrated to React
react-bootstrap.
Also for a complete project integrated with Material Design is here
React - Material Design for Bootstrap
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,698 |
{"url":"http:\/\/openstudy.com\/updates\/4f54014ee4b01ed613831793","text":"## sasogeek 3 years ago A solid weighs W1 in air, W2 when fully immersed in water, and W3 when fully immersed in another liquid. The relative density of the liquid is A. W1\/(W2-W3) B. W1\/(W1-W2) C. (W1-W3)\/(W1-W2) D(W1-W2)\/(W1-W3) I'm thinking the answer is C but I'm not quite sure... relative density is density of substance divided by density of water... A and B are eliminated since we're looking for relative density of the liquid. and with C and D, C is the one with W2 in the denominator... :\/ what's the right answer?\n\n1. JamesJ\n\nWrite down the equation of the forces acting on the mass in water in terms of the objects mass, volume and and density of water; and the variables W1 and W2 Then write down the equation for the section situation in terms of the same variables, except W3 for W2.\n\n2. JamesJ\n\nThen you want to find the expression for $\\frac{\\rho_{liquid}}{\\rho_{water}}$\n\n3. sasogeek\n\nI'll try that, thanks\n\n4. JamesJ\n\n(and in second situation density of liquid for density of water, obviously)","date":"2015-04-18 16:53:15","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\": 0.7856147289276123, \"perplexity\": 926.8861416692965}, \"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-2015-18\/segments\/1429246635639.11\/warc\/CC-MAIN-20150417045715-00246-ip-10-235-10-82.ec2.internal.warc.gz\"}"} | null | null |
package com.annimon.stream.function;
import com.annimon.stream.Objects;
import java.util.Arrays;
import org.jetbrains.annotations.NotNull;
/**
* Represents a predicate (function with boolean type result).
*
* @param <T> the type of the input to the function
*/
public interface Predicate<T> {
/**
* Tests the value for satisfying predicate.
*
* @param value the value to be tested
* @return {@code true} if the value matches the predicate, otherwise {@code false}
*/
boolean test(T value);
class Util {
private Util() { }
/**
* Applies logical AND to predicates.
*
* @param <T> the type of the input to the function
* @param p1 the first predicate
* @param p2 the second predicate
* @return a composed {@code Predicate}
* @throws NullPointerException if {@code p1} or {@code p2} is null
*/
public static <T> Predicate<T> and(
@NotNull final Predicate<? super T> p1,
@NotNull final Predicate<? super T> p2) {
Objects.requireNonNull(p1, "predicate1");
Objects.requireNonNull(p2, "predicate2");
return new Predicate<T>() {
@Override
public boolean test(T value) {
return p1.test(value) && p2.test(value);
}
};
}
/**
* Applies logical AND to multiple predicates.
*
* @param <T> the type of the input to the function
* @param p1 the first predicate
* @param p2 the second predicate
* @param rest the rest predicates
* @return a composed {@code Predicate}
* @throws NullPointerException if any of predicates are null
* @since 1.2.1
*/
public static <T> Predicate<T> and(
@NotNull final Predicate<? super T> p1,
@NotNull final Predicate<? super T> p2,
@NotNull final Predicate<? super T>... rest) {
Objects.requireNonNull(p1, "predicate1");
Objects.requireNonNull(p2, "predicate2");
Objects.requireNonNull(rest, "rest");
Objects.requireNonNullElements(Arrays.asList(rest));
return new Predicate<T>() {
@Override
public boolean test(T value) {
boolean result = p1.test(value) && p2.test(value);
if (!result) return false;
for (Predicate<? super T> p : rest) {
if (!p.test(value)) return false;
}
return true;
}
};
}
/**
* Applies logical OR to predicates.
*
* @param <T> the type of the input to the function
* @param p1 the first predicate
* @param p2 the second predicate
* @return a composed {@code Predicate}
* @throws NullPointerException if {@code p1} or {@code p2} is null
*/
public static <T> Predicate<T> or(
@NotNull final Predicate<? super T> p1,
@NotNull final Predicate<? super T> p2) {
Objects.requireNonNull(p1, "predicate1");
Objects.requireNonNull(p2, "predicate2");
return new Predicate<T>() {
@Override
public boolean test(T value) {
return p1.test(value) || p2.test(value);
}
};
}
/**
* Applies logical OR to multiple predicates.
*
* @param <T> the type of the input to the function
* @param p1 the first predicate
* @param p2 the second predicate
* @param rest the rest predicates
* @return a composed {@code Predicate}
* @throws NullPointerException if any of predicates are null
*/
public static <T> Predicate<T> or(
@NotNull final Predicate<? super T> p1,
@NotNull final Predicate<? super T> p2,
@NotNull final Predicate<? super T>... rest) {
Objects.requireNonNull(p1, "predicate1");
Objects.requireNonNull(p2, "predicate2");
Objects.requireNonNull(rest, "rest");
Objects.requireNonNullElements(Arrays.asList(rest));
return new Predicate<T>() {
@Override
public boolean test(T value) {
boolean result = p1.test(value) || p2.test(value);
if (result) return true;
for (Predicate<? super T> p : rest) {
if (p.test(value)) return true;
}
return false;
}
};
}
/**
* Applies logical XOR to predicates.
*
* @param <T> the type of the input to the function
* @param p1 the first predicate
* @param p2 the second predicate
* @return a composed {@code Predicate}
* @throws NullPointerException if {@code p1} or {@code p2} is null
*/
public static <T> Predicate<T> xor(
@NotNull final Predicate<? super T> p1,
@NotNull final Predicate<? super T> p2) {
Objects.requireNonNull(p1, "predicate1");
Objects.requireNonNull(p2, "predicate2");
return new Predicate<T>() {
@Override
public boolean test(T value) {
return p1.test(value) ^ p2.test(value);
}
};
}
/**
* Applies logical negation to predicate.
*
* @param <T> the type of the input to the function
* @param predicate the predicate to be negated
* @return a composed {@code Predicate}
* @throws NullPointerException if {@code p1} is null
*/
public static <T> Predicate<T> negate(@NotNull final Predicate<? super T> predicate) {
Objects.requireNonNull(predicate);
return new Predicate<T>() {
@Override
public boolean test(T value) {
return !predicate.test(value);
}
};
}
/**
* Checks that input value is not null.
*
* @param <T> the type of the input to the function
* @return {@code Predicate} that checks value to be not null
*/
public static <T> Predicate<T> notNull() {
return new Predicate<T>() {
@Override
public boolean test(T value) {
return value != null;
}
};
}
/**
* Creates a safe {@code Predicate}.
*
* @param <T> the type of the input to the function
* @param throwablePredicate the predicate that may throw an exception
* @return a {@code Predicate} or {@code false} if exception was thrown
*/
public static <T> Predicate<T> safe(
@NotNull ThrowablePredicate<? super T, Throwable> throwablePredicate) {
return safe(throwablePredicate, false);
}
/**
* Creates a safe {@code Predicate}.
*
* @param <T> the type of the input to the function
* @param throwablePredicate the predicate that may throw an exception
* @param resultIfFailed the result which returned if exception was thrown
* @return a {@code Predicate} or {@code resultIfFailed}
* @throws NullPointerException if {@code throwablePredicate} is null
*/
public static <T> Predicate<T> safe(
@NotNull final ThrowablePredicate<? super T, Throwable> throwablePredicate,
final boolean resultIfFailed) {
Objects.requireNonNull(throwablePredicate);
return new Predicate<T>() {
@Override
public boolean test(T value) {
try {
return throwablePredicate.test(value);
} catch (Throwable throwable) {
return resultIfFailed;
}
}
};
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 349 |
#ifndef _CRIS_PGALLOC_H
#define _CRIS_PGALLOC_H
#include <linux/threads.h>
#include <linux/mm.h>
#define pmd_populate_kernel(mm, pmd, pte) pmd_set(pmd, pte)
#define pmd_populate(mm, pmd, pte) pmd_set(pmd, page_address(pte))
#define pmd_pgtable(pmd) pmd_page(pmd)
/*
* Allocate and free page tables.
*/
static inline pgd_t *pgd_alloc (struct mm_struct *mm)
{
return (pgd_t *)get_zeroed_page(GFP_KERNEL);
}
static inline void pgd_free(struct mm_struct *mm, pgd_t *pgd)
{
free_page((unsigned long)pgd);
}
static inline pte_t *pte_alloc_one_kernel(struct mm_struct *mm, unsigned long address)
{
pte_t *pte = (pte_t *)__get_free_page(GFP_KERNEL|__GFP_ZERO);
return pte;
}
static inline pgtable_t pte_alloc_one(struct mm_struct *mm, unsigned long address)
{
struct page *pte;
pte = alloc_pages(GFP_KERNEL|__GFP_ZERO, 0);
if (!pte)
return NULL;
if (!pgtable_page_ctor(pte)) {
__free_page(pte);
return NULL;
}
return pte;
}
static inline void pte_free_kernel(struct mm_struct *mm, pte_t *pte)
{
free_page((unsigned long)pte);
}
static inline void pte_free(struct mm_struct *mm, pgtable_t pte)
{
pgtable_page_dtor(pte);
__free_page(pte);
}
#define __pte_free_tlb(tlb,pte,address) \
do { \
pgtable_page_dtor(pte); \
tlb_remove_page((tlb), pte); \
} while (0)
#define check_pgt_cache() do { } while (0)
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,355 |
\section{Introduction and Results}
Let ${\sf Mat}_{n\times n}$ be the space of $n\times n$ matrices over a field $\Bbbk$; the coordinate ring is
$R={\Bbbk}[x_{ij}]_{1\leq i,j\leq n}$. Let $GL_n$ be the general linear group of invertible $n\times n$ matrices with a
Borel subgroup $B$ of upper triangular matrices and $B_-$ of lower triangular matrices. Let $B_-\times B$ act on
${\sf Mat}_{n\times n}$ by $(b_-,b)\cdot M= b_{-}Mb^{-1}$. Let $w$ be a permutation in the symmetric group ${\mathfrak S}_n$ on
$[n]=\{1,2,\ldots,n\}$, and suppose $M_w$ is its permutation matrix with $1$ in position $(i,w(i))$ and $0$'s elsewhere.
\begin{definition}[\cite{Fulton,KM:annals}]
The \emph{matrix Schubert variety} $X_w$ is the $B_{-}\times B$ orbit closure of $M_w$.
\end{definition}
\begin{definition}[\cite{Fulton,KM:annals}]
The \emph{Schubert determinantal ideal} $I_w\subset R$ is the defining ideal of $X_w$.
\end{definition}
Since \cite{Fulton}, there has been interest in matrix Schubert varieties and the Schubert determinantal ideals; see, \emph{e.g.,} \cite{KM:annals, KMY, Hsiao, Fink, Hamaker, Klein, Klein.Weigandt, Rajchgot, Pechenik} and references therein.
Let $r_{ij}=r_{ij}(w)$ be the \emph{rank function} of $w$. It counts the number of $1$'s weakly northwest of position $(i,j)$ in $M_w$. Let $M^{[i,j]}$ denote the
northwest $i\times j$ submatrix of a generic matrix $M\in {\sf Mat}_{n\times n}$. In \cite{Fulton}, it is proved that
$I_w$ is indeed generated by determinants:
\begin{equation}
\label{eqn:generators}
I_w=\langle (r_{ij}+1)\times (r_{ij}+1) \text{ size minors of $M^{[i,j]}$}, (i,j)\in [n]^2\rangle,
\end{equation}
and that this ideal is prime \cite[Corollary~3.13]{Fulton}.
In \emph{loc.~cit.}, W.~Fulton minimizes the \emph{description} of the generators (\ref{eqn:generators}).
The \emph{Rothe diagram} of $w$ is
\[D(w)=\{(i,j)\in [n]^2: i>w(j), j<w^{-1}(i)\}.\]
\emph{Fulton's essential set} is
\[E(w)=\{(i,j)\in D(w): (i+1,j),(i,j+1)\not\in D(w)\}.\]
W.~Fulton proved that $I_w=\langle (r_{ij}+1)\times (r_{ij}+1) \text{\ size minors of $M^{[i,j]}$}, (i,j)\in E(w)\rangle$.
This is a minimal list of rank conditions needed to describe $I_w$ but
does not nontrivially shorten the \emph{set} of generators. The minors in this description are called the \emph{essential minors} of $I_w$.
\begin{example}[Essential minors do not form a minimal generating set] The reader can check
\[I_{3142}=\left\langle x_{11}, x_{12},
\left|\begin{matrix} x_{21} & x_{22}\\ x_{31} & x_{32}\end{matrix}\right|,
\left|\begin{matrix} x_{11} & x_{12}\\ x_{21} & x_{22}\end{matrix}\right|,
\left|\begin{matrix} x_{11} & x_{12}\\ x_{31} & x_{32}\end{matrix}\right|\right\rangle.\]
The latter two essential minors can be dispensed with; they are implied by the first two.
\end{example}
For $I,J\subseteq [n]$ with $|I| = |J|$, define $m_{I,J}$
to be the determinant of the submatrix of $M$ with row and column indices $I$ and $J$ respectively.
An essential generator $m_{I,J}$
\emph{belongs to} $(i,j)\in E(w)$ if $I\subseteq [i],J\subseteq[j]$ and $r = r_{ij}+1$.
\begin{definition}
A minor $m_{I,J}$
\emph{attends} $M^{[i',j']}$ if $|I\cap[i']|> r_{i',j'}$ and $J\cap[j']= r_{ij}+1$ or
$|I\cap[i']|=r_{i,j}+1$ and $J\cap[j']> r_{i',j'}$.
\end{definition}
\begin{definition}
A minor $m_{I,J}$ that belongs to $(i,j)\in E(w)$ is \emph{elusive} if it does not attend $M^{[i',j]}$ for all
$(i',j')\in E(w)$ such that $r_{i',j}<r_{i,j}$.
\end{definition}
\begin{theorem}
\label{thm:main}
$I_w$ is minimally generated by elusive minors. Moreover, for any $b\in D(w)$
there exists an elusive minor whose southeast corner is $b$.
\end{theorem}
\begin{example}\label{exa:enumerate}
Let $w=619723458$. Theorem~\ref{thm:main} is a handy way to hand compute the size of a minimal
generating set. Here, the minimal generating set
contains $5$ generators of degree $1$, ${3 \choose 2}{5\choose 2}$ generators of degree $2$ and $1+{5\choose 1}{3 \choose 2}$
generators of degree $3$.
An exercise is $w=13865742$ \cite[Example~1.3.5]{KM:annals}. A minimum generating set is of size $104$ consists of $21$ many $2\times 2$ minors and $83$
many $3\times 3$ minors.\footnote{This example is also considered in the unpublished Section~3 of earlier (v1, v2) {\sf arXiv} preprint versions of \cite{KM:annals}. The notion of \emph{attends} is more general than ``\emph{causes}'' used in those preprints and the published version.}
\end{example}
A.~Knutson-E.~Miller \cite[Theorem~B]{KM:annals} proves that the essential minors of $I_w$ form a Gr\"obner basis with respect to any \emph{antidiagonal term order} $\prec$, that
is, a monomial ordering that picks the antidiagonal term of an minor (one example is the lexicographic ordering obtained by reading the rows of the generic matrix right to left in rows, and from top to bottom).
A.~Knutson, E.~Miller and the second author \cite[Theorem~3.8]{KMY} prove the same result for \emph{diagonal term order}
$\prec'$, but under the hypothesis that $w$ is \emph{vexillary} (that is $2143$-avoiding; see the definition of pattern
avoidance below). We refine these statements:
\begin{corollary}
\label{cor:Grobner}
The set of elusive minors is a Gr\"obner basis for $I_w$ under an antidiagonal term order $\prec$.
If we assume $w$ is vexillary, the same statement is true under diagonal term order $\prec'$.
That is, in either case, $I_w$ has a Gr\"obner basis given by a set of
minimal generators.
\end{corollary}
The codimension of $X_w\subseteq \Bbbk^{n^2}$ is $\ell(w)=\#D(w)$, that is, the number of inversions of $w$
\cite[Corollary~3.13]{Fulton}. Since the size of a minimal generating set is an invariant, $X_w$
is a complete intersection if and only if the size of the set of its elusive minors is $\ell(w)$. Using this, we give a self-contained proof of the result below of H.~Ulfarsson-A.~Woo \cite[Corollary~6.3]{Woo}, which is a pattern avoidance characterization of matrix Schubert varieties that are complete intersections. Their result came after an earlier characterization by J.~C.~Hsiao \cite[Theorem~5.2]{Hsiao} which depends on the Gr\"obner basis theorem of \cite[Theorem~B]{KM:annals}.
Recall $w\in {\mathfrak S}_n$ \emph{pattern includes} $u\in S_{m}$ if there exist indices $i_1<i_2<\ldots<i_m$
such that $w(i_1),\ldots, w(i_m)$ is in the same relative order as $u(1),\ldots,u(m)$. Furthermore, $w$ avoids
$u$ if no such indices exist.
\begin{corollary}[\cite{Hsiao, Woo}]\label{cor:lci}
$X_w$ is a complete intersection if and only if $w$ avoids $1342, 1432, 1423$.\footnote{In \cite[Corollary~6.3]{Woo}, the additional patterns $31524,24153$ and $351624$ are listed. However these follow from the size $4$ pattterns.}
\end{corollary}
In fact, the proofs of \cite{Hsiao, Woo} construct a minimal set of generators for $I_w$ to prove ``$\Leftarrow$''. However, their arguments do not do this outside of those cases.
\section{Proofs}
\subsection{Proof of Theorem~\ref{thm:main}}
Suppose a minor $m$ belonging to $(i,j)\in E(w)$ is not elusive; say it attends $M^{[i',j']}$ for $(i',j')\in E(w)$ satisfying $r_{i',j'}(w)<r_{i,j}(w)$. Then it follows by induction using cofactor expansion that $m$ is in the ideal
generated by the $(r_{i',j'}+1)\times (r_{i',j'}+1)$ minors belonging to $M^{[i',j']}$. Hence $m$ can be dispensed with.
Conversely, suppose $m=m_{I,J}$ belonging to $e=(i,j)\in E(w)$ is elusive.
Let $I=\{1\leq i_1<i_{2}<\ldots<i_{r+1}\leq i\}$ and $J=\{1\leq j_1<j_2<\ldots<j_{r+1}\leq j\}$ where $r=r_{i,j}(w)$.
\begin{claim}\label{claim:rank}
For any $1\leq k<r+1$, we have $r_{i_k,j}\geq k$ and $r_{i,j_k}\geq k$.
\end{claim}
\noindent \emph{Proof of Claim~\ref{claim:rank}:} We will prove $r_{i_k,j}\geq k$; $r_{i,j_k}\geq k$ follows from the same reasoning.
We proceed by induction. For $k=1$, suppose $r_{i_1,j} = 0 <k$, then $(i_1,j)\in D(w)$. There is $(a,b)\in E(w)$
weakly southeast of $(i_1,j)$ and in the same connected component of $D(w)$. Since $r_{a,b} = r_{i_1,j} = 0$, $b\geq j$ and $I\cap[a]\geq 1>0$, $m_{I,J}$ attends $M^{[a,b]}$, a contradiction.
Now suppose $r_{i_k,j}\geq k$ for all $1\leq k<s$ for some $s<r+1$. If $(i_{s},j)\in D(w)$, there is a $(a,b)\in E(w)$ that is in the same connected component as $(i_{s},j)$ and weakly southeast of $(i_{s},j)$. If $r_{i_{s},j} = s-1$, then $m_{I,J}$ attends $M^{[a,b]}$, a contradiction. So $r_{i_{s},j}\geq s$ in this case. Now if $(i_{s},j)\notin D(w)$, since $(i,j)\in D(w)$ and $i>i_{s}$, we know that $w(i_{s}) < j$. Since $r_{{i_{s-1}},j}\geq s-1$, we see $r_{i_{s},j}\geq r_{i_{s-1},j}+1 = s$, completing the induction step.
\qed
To prove that $m=m_{I,J}$ is necessary as a generator, it suffices
to find a point $P\in {\sf Mat}_{n\times n}$ such that $m$ does not vanish at $P$ but every other essential generator does vanish.
Set
$P_{a,b}=1$ if $a=i_t$ and $b=j_{r-t+2}$ for $1\leq t\leq r+1$, and let all other entries be $0$. In words, $P$ places $1$'s on the antidiagonal of $m$. Evidently $m$ does not vanish at $P$.
It remains to prove all other essential minors do vanish at $P$. Suppose, to the contrary that
$m'$ is a minor that belongs to $e'\in E(w)$ but does not vanish at $P$. Clearly $r_{e'}(w)<r_e(w)$. Let $e' = (i',j')$. If $e'$ is not in the rectangle with corners $(1,1)$ and $(i_{r+1}-1,j_{r+1}-1)$,
by definition, $m$ attends $M^{[e']}$ contradicting the assumption that $m$ is elusive.
Thus, the only possibility is that $i'<i_{r+1}$ and $j'<j_{r+1}$ as depicted below.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale=0.45]
\fill[color=black!40, fill=yellow, very thick] (7,8) rectangle (6,9);
\fill[color=black!40, fill=yellow, very thick] (4,6) rectangle (3,7);
\fill[color=black!40, fill=yellow, very thick] (2,3) rectangle (1,4);
\draw[black, thick] (0,0) -- (10,0) -- (10,10) -- (0,10) -- (0,0);
\draw[black, thick] (7,1) -- (8,1) -- (8,2) -- (7,2) -- (7,1);
\draw[black, thick] (5,4) -- (6,4) -- (6,5) -- (5,5) -- (5,4);
\draw[red, thick, dashed] (0,8) -- (8,8);
\draw[red, thick, dashed] (2,10) -- (2,1);
\draw[blue, thick] (0,6) -- (8,6);
\draw[blue, thick] (4,10) -- (4,1);
\draw (7.5,1.5) node{$e$};
\draw (5.5,4.5) node{$e'$};
\draw (-0.35, 1.5) node{$i$};
\draw (-0.35, 3.5) node{$i_3$};
\draw (-0.35, 6.5) node{$i_2$};
\draw (-0.35, 8.5) node{$i_1$};
\draw (-0.35, 4.5) node{$i'$};
\draw (7.5, 10.45) node{$j$};
\draw (6.5, 10.45) node{$j_3$};
\draw (3.5, 10.45) node{$j_2$};
\draw (1.5, 10.45) node{$j_1$};
\draw (5.5, 10.45) node{$j'$};
\draw (6.5, 8.5) node{$1$};
\draw (3.5, 6.5) node{$1$};
\draw (1.5, 3.5) node{$1$};
\draw[gray, dashed] (0,1) -- (7,1);
\draw[gray, dashed] (8,10) -- (8,2);
\end{tikzpicture}
\label{fig:Jan9aaa}
\end{center}
\end{figure}
Let us assume that $i_p\leq i'<i_{p+1}$ and $j_{\ell}\leq j'<j_{\ell+1}$ for some $0\leq p,\ell<r+1$, where $i_0 = j_0: = 1$
(in the figure, $p=\ell=2$). Since $m'$ does not vanish at $P$, it is straightforward that $p+\ell>r+1$
(otherwise, $m'$ only involves $0$ entries)
and $r_{i',j'}<p+\ell-(r+1)$ (the right hand is the number of $1$'s that appear in the northwest $i_p\times j_{\ell}$
rectangle of $M_w$).
In particular, this implies that
\[r_{i_p,j_\ell}<p+\ell-(r+1).\]
By Claim~\ref{claim:rank}, $r_{i,j_\ell}\geq \ell$. Since $r_{i,j} = r$, we obtain
\[r_{i_p,j}-r_{i_p,j_\ell}\leq r_{i,j}-r_{i,j_\ell} \leq r-\ell.\]
Hence,
\[r_{i_p,j}<p+\ell-(r+1)+r-\ell = p-1.\]
Yet by Claim~\ref{claim:rank}, $r_{i_p,j}\geq p$, a contradiction. So all other essential minors vanish on $P$.
We now turn to the second statement of the theorem, restated here in more exact form:
\begin{proposition}\label{prop:elusiveSE}
For $b \!=\! (i,j)\!\in\! D(w)$, $m_{[i-r,i],[j-r,j]}$ is an elusive minor with southeast corner $b$.
\end{proposition}
\begin{proof}
Let $r = r_b(w), I = [i-r,i]$ and $J = [j-r,j]$. We will show that $m_{I,J}$ is an elusive minor. Since $r_e(w) = r$ for any $e\in E(w)$ that is in the same connected component as $b$ in $D(w)$, $m_{I,J}$ belongs to any such $e$. Suppose $m_{I,J}$ attends $M^{[e']}$ for some $e' = (i',j')\in E(w)$. We can assume, without loss, that $i'< i$
and $j'>j$.
Since $b,e'\in D(w)$, we know that $(i',j)\in D(w)$. Let $k = i-i'$, we then have
\[r_{i',j}(w)\geq r-k+1.\]
Since $r_{e'}(w)\geq r_{i',j}(w)$,
\[r_{e'}(w)\geq r-k+1.\]
Thus any minor that belongs to $e'$ has size at least $r-k+2$. Since
\[|I\cap[i']| = r-k+1 <r-k+2,\]
$m_{I,J}$ does not attend $M^{[e']}$, a contradiction. Therefore $m_{I,J}$ is elusive, as claimed.
\end{proof}
\subsection{Proof of Corollary~\ref{cor:Grobner}}
For $f\in R$, let ${\sf init}_{\prec}(f)$ be the initial term of $f$ under $\prec$. By definition, a generating set ${\mathcal S}$ of $I_w$ is a Gr\"obner basis if
\[\langle {\sf init}_{\prec}(f): f\in {\mathcal S}\rangle=
\langle {\sf init}_{\prec}(f): f\in I_w\rangle.\]
By \cite[Theorem~B]{KM:annals}, the essential minors are a Gr\"{o}bner basis under $\prec$.
Therefore it suffices to show that if $m$ is a non-elusive minor then
${\sf init}_{\prec} \, m$ is divisible by ${\sf init}_{\prec} \, m'$ where $m'$ is an elusive minor. We proceed by induction, ordering the essential set by rank value. In the base case where the rank is $0$, all the associated
$1\times 1$ minors are elusive, trivially. Suppose $m$ belongs to $(i,j)\in E(w)$ but is not elusive.
Since $m=m_{I,J}$ is not elusive we may suppose, without loss of generality, that there is
$(i',j')\in E(w)$ with $r_{i',j'}(w)<r_{i,j}(w)$
such that $|I\cap[i']|> r_{i',j'}$ and $J\cap[j']= r_{i,j}+1$ (the argument in the other case is similar).
Thus, there is a minor $m_{I',J'}$ that belongs to $(i',j')$ (and of size $(r_{i',j'}+1)\times (r_{i',j'}+1)$) whose
antidiagonal term divides that of $m_{I,J}$. More precisely $I'$ consists of the $r_{i',j'}+1$ smallest indices of $I$, and $J'$ consists of the $r_{i',j'}+1$ largest indices of $J$. If $m_{I',J'}$ is elusive we are done. Otherwise by induction
${\sf init}_{\prec} \, m_{I',J'}$ is divisible by ${\sf init}_{\prec} \, m_{I'',J''}$ for some elusive $m_{I'',J''}$ in which case
${\sf init}_{\prec} m_{I'',J''}$ divides ${\sf init}_{\prec} m_{I,J}$, as desired.
If $w$ is vexillary and
$e,e'\in E(w)$ then $e$ cannot be strictly northwest of $e'$. Using this and the Gr\"obner basis result \cite[Theorem~3.8]{KMY}, the second sentence follows like the first.
The final sentence of the statement then is immediate from Theorem~\ref{thm:main}.
\subsection{Proof of Corollary~\ref{cor:lci}}
\begin{lemma}
\label{lemma:firstone}
$X_w$ is a complete intersection if and only if $X_{w^{-1}}$ is a complete intersection.
\end{lemma}
\begin{proof}
Follows from (\ref{eqn:generators}).
\end{proof}
\begin{lemma}[Shifting]\label{shifting}
If $m_{I,J}$ is an elusive minor of $I_w$ then $m_{I',J'}$ is elusive whenever $I'=(I-\{i\})\cup \{t\}$ where $t>i$ and
$t\not\in I$ or similarly for $J'$.
\end{lemma}
\begin{proof}
Immediate from the definitions.
\end{proof}
$(\Rightarrow)$: We prove the contrapositive. Suppose $w$ pattern embeds $1342$ or $1432$. Let $a_1<a_2<a_3<a_4$ be such that $w(a_1)<w(a_4)<w(a_2),w(a_3)$. Set $b_1 = w(a_1), b_2 = w(a_4), b_3 = w(a_2), b_4 = w(a_3)$, then $e = (a_3,b_2)\in D(w)$ as shown here:
\begin{figure}[h]
\begin{center}
\begin{subfigure}{0.45\textwidth}
\hspace{1cm}
\begin{tikzpicture}[scale = 0.40]
\draw[black, thick] (0,0) -- (10,0) -- (10,10) -- (0,10) -- (0,0);
\draw[black, thick] (3,6) -- (3,7) -- (4,7) -- (4,6) -- (3,6) ;
\draw[black, thick] (3,4) -- (4,4) -- (4,5) -- (3,5) -- (3,4) ;
\draw[black, thick] (5.5,6.5) -- (5.5,0);
\draw[black, thick] (5.5,6.5) -- (10,6.5);
\draw[black, thick] (1.5,8.5) -- (1.5,0);
\draw[black, thick] (1.5,8.5) -- (10,8.5);
\draw[black, thick] (3.5,1.5) -- (3.5,0);
\draw[black, thick] (3.5,1.5) -- (10,1.5);
\draw[black, thick] (7.5,4.5) -- (7.5,0);
\draw[black, thick] (7.5,4.5) -- (10,4.5);
\draw (3.5,4.5) node{$e$};
\draw (1.5,8.5) node{$\bullet$};
\draw (5.5,6.5) node{$\bullet$};
\draw (7.5,4.5) node{$\bullet$};
\draw (3.5,1.5) node{$\bullet$};
\draw (-0.55, 1.5) node{$a_4$};
\draw (-0.55, 6.5) node{$a_2$};
\draw (-0.55, 8.5) node{$a_1$};
\draw (-0.55, 4.5) node{$a_3$};
\draw (7.5, 10.55) node{$b_4$};
\draw (3.5, 10.55) node{$b_2$};
\draw (1.5, 10.55) node{$b_1$};
\draw (5.5, 10.55) node{$b_3$};
\end{tikzpicture}
\caption{\normalsize{$w$ embeds $1342$}}
\label{fig:1342}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\hspace{1cm}
\begin{tikzpicture}[scale=0.40]
\draw[black, thick] (0,0) -- (10,0) -- (10,10) -- (0,10) -- (0,0);
\draw[black, thick] (3,6) -- (3,7) -- (4,7) -- (4,6) -- (3,6) ;
\draw[black, thick] (3,4) -- (4,4) -- (4,5) -- (3,5) -- (3,4) ;
\draw[black, thick] (7.5,6.5) -- (7.5,0);
\draw[black, thick] (7.5,6.5) -- (10,6.5);
\draw[black, thick] (1.5,8.5) -- (1.5,0);
\draw[black, thick] (1.5,8.5) -- (10,8.5);
\draw[black, thick] (3.5,1.5) -- (3.5,0);
\draw[black, thick] (3.5,1.5) -- (10,1.5);
\draw[black, thick] (5.5,4.5) -- (5.5,0);
\draw[black, thick] (5.5,4.5) -- (10,4.5);
\draw (3.5,4.5) node{$e$};
\draw (1.5,8.5) node{$\bullet$};
\draw (7.5,6.5) node{$\bullet$};
\draw (5.5,4.5) node{$\bullet$};
\draw (3.5,1.5) node{$\bullet$};
\draw (-0.55, 1.5) node{$a_4$};
\draw (-0.55, 6.5) node{$a_2$};
\draw (-0.55, 8.5) node{$a_1$};
\draw (-0.55, 4.5) node{$a_3$};
\draw (7.5, 10.55) node{$b_4$};
\draw (3.5, 10.55) node{$b_2$};
\draw (1.5, 10.55) node{$b_1$};
\draw (5.5, 10.55) node{$b_3$};
\end{tikzpicture}
\caption{\normalsize{$w$ embeds $1432$}}
\label{fig:1432}
\end{subfigure}
\label{}
\end{center}
\end{figure}
Furthermore, we can assume, without loss, that for any $a<a_1$, we have $w(a)>b_2$. That is, we can pick
$(a_1,b_1)$ to be the highest non-zero entry in $M_w$ that is strictly northwest of $e$. Since $w(a_2) = b_3>b_2$, we have $a_3>r_e(w)+1$. Set $r = r_e(w)$ and let
\[I = [a_3-r,a_3], I' = \{a_3-r-1\}\cup[a_3-r+1,a_3], J = [b_2-r,b_2].\]
Since $m_{I,J}$ is elusive (by Proposition~\ref{prop:elusiveSE}), to show that $m_{I',J}$ is also elusive, it is enough to prove that there does not exist $(i',j')\in E(w)$ such that $J\cap[j'] = J$, $r_{i',j'} = 0$ and $I\cap[i'] = \{a_3-r-1\}$.
Suppose not. Indeed, since $w(a_2)>b_2$, we know that $a_3-r-1\geq a_1$ and thus $r_{i',j'}\geq 1$,
a contradiction. Therefore, there are at least two elusive minors whose southeast corner is $e$ and therefore, by Theorem~\ref{thm:main},
$X_w$ is not a complete intersection.
Since $w$ includes $1342$ if and only if $w^{-1}$ includes $1423$, we are done by Lemma~\ref{lemma:firstone}.
$(\Leftarrow)$: We again argue the contrapositive. Suppose $X_w$ is not a complete intersection. By Theorem~\ref{thm:main} there is either
$e = (i,j)\in D(w)$ having more than one elusive minor $m$ with southeast corner $e$, or there is an $e=(i,j)\in [n]^2-D(w)$
that is the southeast corner of an elusive minor $m'$ that belongs to $e'\in E(w)$. In the second case, by
using Lemma~\ref{shifting} to repeatedly shift the southmost row and/or eastmost column used by
$m'$ one obtains another elusive minor $m''$ with southeast corner $e'$ which
is different than the elusive minor from the proof of Proposition~\ref{prop:elusiveSE}. Hence we assume we are in the first
case.
Let $r = r_e(w)$. By using Lemma~\ref{lemma:firstone} or Lemma~\ref{shifting}, we may assume that $m_{I',J}$ is an elusive minor where
\[I' = \{i-r-1\}\cup[i-r+1,i], J = [j-r,j].\]
Since $m_{I',J}$ is elusive, $r_{i-r-1,j}\geq 1$. Also, since $(i,j)\in D(w)$, wither $w(i-r-1)<j$ or $(i-r-1,j)\in D(w)$ is true.
Suppose $w(i-r-1)<j$, by the pigeonhole principle, there exists $a$ such that $(a,j)\in D(w)$ and $i-r-1<a<i$. As a result,
\[i-r-1<a<i<w^{-1}(j)\ \text{and}\ w(i-r-1)<j<w(a),w(i).\]
Therefore $w$ embeds $1342$ or $1432$, and we are done.
Hence $(i-r-1,j)\in D(w)$. Since $r_{i-r-1,j}\geq 1$, there exists $a<i-r-1$ such that $w(a)<j$. Since $(i-r-1,j)\in D(w)$, we get $w(i-r-1)>j$. We then have
\[a<i-r-1<i<w^{-1}(j)\ \text{and }w(a)<j<w(i-r-1),w(i).\]
Therefore $w$ embeds $1342$ or $1432$. \qed
\section*{Acknowledgements}
The authors were supported by an NSF RTG in Combinatorics. Additionally, SG was supported by an NSF graduate fellowship and AY by a Simons Collaboration Grant.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 582 |
{"url":"https:\/\/www.idexlab.com\/openisme\/topic-serendipity\/","text":"Serendipity\n\n14,000,000 Leading Edge Experts on the ideXlab platform\n\nContact Leading Edge Experts & Companies\n\nThe Experts below are selected from a list of 8019 Experts worldwide ranked by ideXlab platform\n\nAndrew Burtonjones - One of the best experts on this subject based on the ideXlab platform.\n\n\u2022 with a little help from my friends cultivating Serendipity in online shopping environments\nInformation & Management, 2019\nCo-Authors: Camille Grange, Izak Benbasat, Andrew Burtonjones\nAbstract:\n\nAbstract Many important findings and discoveries in science and everyday life are the result of Serendipity, that is, the unanticipated occurrence of happy events such as the finding of valuable information. Consumers are increasingly seeking Serendipity in online shopping, where information clutter and preprogramed recommendation systems can make product choice frustrating or mundane. However, it is notoriously difficult to design online shopping environments that induce it. In this study, we explore how social media affordances such as obtaining access to peer-generated content and being connected to online friends can help create the right conditions for Serendipity in online shopping. We supplement this analysis with an account of two individual factors that are also likely to be instrumental in a shopping context, namely, the intensity of shoppers\u2019 information search and their aversion to risk when faced with a product choice. Our investigation relies on a conceptualization of Serendipity that has two defining elements: unexpectedness and informational value. The results of an experimental study in which we manipulated an online product search environment reveal the superiority of designs that incorporate online friendships, and these results support the positive effects of search effort and risk aversion on Serendipity. This study contributes by developing a theoretical framework for the analysis of Serendipity and by explaining how social commerce, that is, the integration of social media and electronic commerce, can cultivate Serendipity.\n\n\u2022 with a little help from my friends cultivating Serendipity in online shopping environments\nSocial Science Research Network, 2017\nCo-Authors: Camille Grange, Izak Benbasat, Andrew Burtonjones\nAbstract:\n\nMany important findings and discoveries from science to everyday life are the result of Serendipity, that is, the occurrence and development of events by chance in a happy or beneficial way. Serendipity is increasingly sought for in the domain of online shopping, where informational clutter and predetermined decision support systems can make product discovery a frustrating endeavor, but it is notoriously difficult to design environments that induce it. In this study, we explore how social media affordances (i.e., obtaining access to peer-generated content and being connected to online friends) might contribute to create the appropriate conditions for Serendipity to manifest, and we supplement this analysis with an account of individual factors that are likely to be instrumental (intensity of shoppers\u2019 search behaviors and shoppers\u2019 risk aversion). Our investigation relies on a conceptualization of Serendipity that has two defining elements: unexpectedness and informational value. The results of an experimental study in which we manipulated a product search environment to test the hypotheses reveal the superiority of designs that incorporate online friendships. Overall, this study contributes by offering a theoretical framework for the study of Serendipity and by explaining how the integration of social media and electronic commerce affordances, and thus, social commerce, can cultivate it.\n\nWei Xue - One of the best experts on this subject based on the ideXlab platform.\n\n\u2022 Serendipity in dark photon searches\nJournal of High Energy Physics, 2018\nCo-Authors: Philip Ilten, Yotam Soreq, Mike Williams, Wei Xue\nAbstract:\n\nSearches for dark photons provide serendipitous discovery potential for other types of vector particles. We develop a framework for recasting dark photon searches to obtain constraints on more general theories, which includes a data-driven method for determining hadronic decay rates. We demonstrate our approach by deriving constraints on a vector that couples to the B-L current, a leptophobic B boson that couples directly to baryon number and to leptons via B - \u03b3 kinetic mixing, and on a vector that mediates a protophobic force. Our approach can easily be generalized to any massive gauge boson with vector couplings to the Standard Model fermions, and software to perform any such recasting is provided at https:\/\/gitlab.com\/philten\/darkcast .\n\n\u2022 Serendipity in dark photon searches\narXiv: High Energy Physics - Phenomenology, 2018\nCo-Authors: Philip Ilten, Yotam Soreq, Mike Williams, Wei Xue\nAbstract:\n\nSearches for dark photons provide serendipitous discovery potential for other types of vector particles. We develop a framework for recasting dark photon searches to obtain constraints on more general theories, which includes a data-driven method for determining hadronic decay rates. We demonstrate our approach by deriving constraints on a vector that couples to the $B\\!-\\!L$ current, a leptophobic $B$ boson that couples directly to baryon number and to leptons via $B$-$\\gamma$ kinetic mixing, and on a vector that mediates a protophobic force. Our approach can easily be generalized to any massive gauge boson with vector couplings to the Standard Model fermions, and software to perform any such recasting is provided at this https URL .\n\nCamille Grange - One of the best experts on this subject based on the ideXlab platform.\n\n\u2022 with a little help from my friends cultivating Serendipity in online shopping environments\nInformation & Management, 2019\nCo-Authors: Camille Grange, Izak Benbasat, Andrew Burtonjones\nAbstract:\n\nAbstract Many important findings and discoveries in science and everyday life are the result of Serendipity, that is, the unanticipated occurrence of happy events such as the finding of valuable information. Consumers are increasingly seeking Serendipity in online shopping, where information clutter and preprogramed recommendation systems can make product choice frustrating or mundane. However, it is notoriously difficult to design online shopping environments that induce it. In this study, we explore how social media affordances such as obtaining access to peer-generated content and being connected to online friends can help create the right conditions for Serendipity in online shopping. We supplement this analysis with an account of two individual factors that are also likely to be instrumental in a shopping context, namely, the intensity of shoppers\u2019 information search and their aversion to risk when faced with a product choice. Our investigation relies on a conceptualization of Serendipity that has two defining elements: unexpectedness and informational value. The results of an experimental study in which we manipulated an online product search environment reveal the superiority of designs that incorporate online friendships, and these results support the positive effects of search effort and risk aversion on Serendipity. This study contributes by developing a theoretical framework for the analysis of Serendipity and by explaining how social commerce, that is, the integration of social media and electronic commerce, can cultivate Serendipity.\n\n\u2022 with a little help from my friends cultivating Serendipity in online shopping environments\nSocial Science Research Network, 2017\nCo-Authors: Camille Grange, Izak Benbasat, Andrew Burtonjones\nAbstract:\n\nMany important findings and discoveries from science to everyday life are the result of Serendipity, that is, the occurrence and development of events by chance in a happy or beneficial way. Serendipity is increasingly sought for in the domain of online shopping, where informational clutter and predetermined decision support systems can make product discovery a frustrating endeavor, but it is notoriously difficult to design environments that induce it. In this study, we explore how social media affordances (i.e., obtaining access to peer-generated content and being connected to online friends) might contribute to create the appropriate conditions for Serendipity to manifest, and we supplement this analysis with an account of individual factors that are likely to be instrumental (intensity of shoppers\u2019 search behaviors and shoppers\u2019 risk aversion). Our investigation relies on a conceptualization of Serendipity that has two defining elements: unexpectedness and informational value. The results of an experimental study in which we manipulated a product search environment to test the hypotheses reveal the superiority of designs that incorporate online friendships. Overall, this study contributes by offering a theoretical framework for the study of Serendipity and by explaining how the integration of social media and electronic commerce affordances, and thus, social commerce, can cultivate it.\n\nGerard Awanou - One of the best experts on this subject based on the ideXlab platform.\n\n\u2022 G.: Finite element differential forms on cubical meshes\n2014\nCo-Authors: Douglas N. Arnold, Gerard Awanou\nAbstract:\n\nAbstract. We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains el-ements of all polynomial degrees and all form degrees. In two dimensions, these include the Serendipity finite elements and the rectangular BDM elements. In three dimensions they include a recent generalization of the Serendipity spaces, and new H(curl) and H(div) finite element spaces. Spaces in the family can be combined to give finite element subcomplexes of the de Rham complex which satisfy the basic hypotheses of the finite element exterior calculus, and hence can be used for stable discretization of a variety of problems. The construction and properties of the spaces are established in a uniform manner using finite element exterior calculus. 1\n\n\u2022 The Serendipity Family of Finite Elements\nFoundations of Computational Mathematics, 2011\nCo-Authors: Douglas N. Arnold, Gerard Awanou\nAbstract:\n\nWe give a new, simple, dimension-independent definition of the Serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s \u2212 r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r . The degrees of freedom are given by moments of degree at most r \u22122 d on each face of dimension d . We establish unisolvence and a geometric decomposition of the space.\n\nSpiros Batas - One of the best experts on this subject based on the ideXlab platform.\n\n\u2022 invs and decoding the three roles of social capital with network closure and structure\nCo-Authors: Spiros Batas, Tanja Leppaaho, Mahmoud Khalik\nAbstract:\n\nThe purpose of this article is to examine how the three different roles of social capital, the efficacy, Serendipity and liability roles interact with network closure and structure. We decode the interrelation between strong, weak, formal, informal and intermediary ties and the three roles of social capital in the internationalization process of nine high technology INVs in Greece and Scotland. We found that the efficacy role of social capital is mainly associated with strong and formal ties, the Serendipity role is related to weak, informal and intermediary ties and finally the liability role is associated with weak and formal ties. We also observed that the efficacious role was the most frequent role, whereas the liability role was the least frequent role.\n\n\u2022 the role of network relationships and social capital in early and later foreign market entries of high technology invs\n2015\nCo-Authors: Spiros Batas, Tanja Leppaaho\nAbstract:\n\nSocial capital and its dynamic nature can offer plausible explanations to the rapid internationalization of International New Ventures (INVs). This paper tries to shed light on the impact of the three roles of social capital, efficacy role, Serendipity role and liability role on international market entries of nine Scottish and Greek high technology INVs. We observed that the efficacious role of social capital occurred more times in both early and later foreign market entries. The liability role had a very limited effect on internationalization. The Serendipity role played an important role in countries with high psychic distance.","date":"2022-07-02 23:49:12","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\": 0.40556520223617554, \"perplexity\": 2179.864790850638}, \"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-2022-27\/segments\/1656104205534.63\/warc\/CC-MAIN-20220702222819-20220703012819-00544.warc.gz\"}"} | null | null |
\section*{\revised{Acknowledgements}}
\looseness=-1
\revised{Special thanks to CloudLab and Chameleon Cloud for making {Zeus\xspace} experiments possible. We would also like to thank the reviewers, our shepherd Jayashree Mohan, and SymbioticLab members for their insightful feedback. We also thank our colleague Rui Liu for his helpful suggestions. This work is in part supported by NSF grants CNS-1909067 and CNS-2104243 and a grant from VMWare. Jae-Won Chung is additionally supported by the Kwanjeong Educational Foundation.}
\section{Job Recurrence in the Alibaba Cluster Trace}\label{sec:appendix-alibaba}
\section{Energy Savings Potential on GPUs}
\label{sec:appendix-eta-potential}
\begin{figure}[!h]
\centering
\subfloat[][NVIDIA A40.]{
\includegraphics[width=0.7\linewidth]{Figures/motivation/eta-potential-all-a40.pdf}
}
\hfil
\subfloat[][NVIDIA V100.]{
\includegraphics[width=0.7\linewidth]{Figures/motivation/eta-potential-all-v100.pdf}
}
\hfil
\subfloat[][NVIDIA RTX6000.]{
\includegraphics[width=0.7\linewidth]{Figures/motivation/eta-potential-all-rtx6000.pdf}
}
\hfil
\subfloat[][NVIDIA P100.]{
\includegraphics[width=0.7\linewidth]{Figures/motivation/eta-potential-all-p100.pdf}
}
\caption{Energy usage normalized against Baseline for DNN training, measured on (a) NVIDIA A40 GPU, (b) NVIDIA V100 GPU, (c) NVIDIA RTX6000 GPU and (d) NVIDIA P100 GPU.}
\label{fig:appendix-eta-potential}
\end{figure}
Figure~\ref{fig:appendix-eta-potential} shows the potential for energy savings on four different generations of NVIDIA GPUs: Ampere (A40), Volta (V100), Turing (RTX6000), and Pascal (P100).
All four generations show that there are sufficient potential for energy savings, motivating {Zeus\xspace}.
\section{TTA vs. ETA for All Workloads}\label{sec:appendix-eta-tta}
\begin{figure}[!t]
\centering
\hfil
\subfloat[][DeepSpeech2]{
\includegraphics[width=0.45\linewidth]{Figures/model/pareto-librispeech.pdf}
}
\hfil
\subfloat[][BERT (QA)]{
\includegraphics[width=0.45\linewidth]{Figures/model/pareto-squad.pdf}
}
\hfil
\subfloat[][BERT (SA)]{
\includegraphics[width=0.45\linewidth]{Figures/model/pareto-sentiment140.pdf}
}
\hfil
\subfloat[][ResNet-50]{
\includegraphics[width=0.45\linewidth]{Figures/model/pareto-imagenet.pdf}
}
\hfil
\subfloat[][ShuffleNet V2]{
\includegraphics[width=0.45\linewidth]{Figures/model/pareto-cifar100.pdf}
}
\hfil
\subfloat[][NeuMF]{
\includegraphics[width=0.45\linewidth]{Figures/model/pareto-movielens-1m.pdf}
}
\hfil
\caption{ETA vs. TTA across all workloads, with Pareto Front and default configuration highlighted. Measured on an NVIDIA V100 GPU.}
\label{fig:appendix-eta-tta-pareto}
\end{figure}
Figure~\ref{fig:appendix-eta-tta-pareto} plots the Pareto Front for all six workloads and the baseline (default batch size and maximum power limit) is shown as a red triangle.
Note that the axes do not start from zero in order to zoom into the Pareto Front.
Data points were gathered on an NVIDIA V100 GPU.
\section{ETA w.r.t. Configurations for All Workloads}\label{sec:appendix-eta-bs-pl}
\begin{figure}[!t]
\centering
\hfil
\subfloat[][DeepSpeech2]{
\includegraphics[width=0.45\linewidth]{Figures/model/bs-eta-librispeech.pdf}
}
\hfil
\subfloat[][BERT (QA)]{
\includegraphics[width=0.45\linewidth]{Figures/model/bs-eta-squad.pdf}
}
\hfil
\subfloat[][BERT (SA)]{
\includegraphics[width=0.45\linewidth]{Figures/model/bs-eta-sentiment140.pdf}
}
\hfil
\subfloat[][ResNet-50]{
\includegraphics[width=0.45\linewidth]{Figures/model/bs-eta-imagenet.pdf}
}
\hfil
\subfloat[][ShuffleNet V2]{
\includegraphics[width=0.45\linewidth]{Figures/model/bs-eta-cifar100.pdf}
}
\hfil
\subfloat[][NeuMF]{
\includegraphics[width=0.45\linewidth]{Figures/model/bs-eta-movielens-1m.pdf}
}
\hfil
\caption{ETA w.r.t batch size of different DNN training workload. The blue shade shows the error margin across repeated runs.}
\label{fig:appendix-bs-eta}
\end{figure}
\begin{figure}[!t]
\centering
\hfil
\subfloat[][DeepSpeech2]{
\includegraphics[width=0.45\linewidth]{Figures/model/pl-eta-librispeech.pdf}
}
\hfil
\subfloat[][BERT (QA)]{
\includegraphics[width=0.45\linewidth]{Figures/model/pl-eta-squad.pdf}
}
\hfil
\subfloat[][BERT (SA)]{
\includegraphics[width=0.45\linewidth]{Figures/model/pl-eta-sentiment140.pdf}
}
\hfil
\subfloat[][ResNet-50]{
\includegraphics[width=0.45\linewidth]{Figures/model/pl-eta-imagenet.pdf}
}
\hfil
\subfloat[][ShuffleNet V2]{
\includegraphics[width=0.45\linewidth]{Figures/model/pl-eta-cifar100.pdf}
}
\hfil
\subfloat[][NeuMF]{
\includegraphics[width=0.45\linewidth]{Figures/model/pl-eta-movielens-1m.pdf}
}
\hfil
\caption{ETA w.r.t GPU power limit of different DNN training workload. Measured on an NVIDIA V100 GPU.}
\label{fig:appendix-pl-eta}
\end{figure}
Figures~\ref{fig:appendix-bs-eta} and~\ref{fig:appendix-pl-eta} respectively show the ETA value when batch size and power limit are swept.
Especially note the convexity of all BS-ETA curves, which justifies the design of our pruning exploration algorithm.
\section{Cumulative Regret of All Workloads}\label{sec:appendix-regret}
\begin{figure}[!t]
\centering
\hfil
\subfloat[][DeepSpeech2]{
\includegraphics[width=0.45\linewidth]{Figures/eval/regret-librispeech-eta-0.5.pdf}
}
\hfil
\subfloat[][BERT (QA)]{
\includegraphics[width=0.45\linewidth]{Figures/eval/regret-squad-eta-0.5.pdf}
}
\hfil
\subfloat[][BERT (SA)]{
\includegraphics[width=0.45\linewidth]{Figures/eval/regret-sentiment140-eta-0.5.pdf}
}
\hfil
\subfloat[][ResNet-50]{
\includegraphics[width=0.45\linewidth]{Figures/eval/regret-imagenet-eta-0.5.pdf}
}
\hfil
\subfloat[][ShuffleNet V2]{
\includegraphics[width=0.45\linewidth]{Figures/eval/regret-cifar100-eta-0.5.pdf}
}
\hfil
\subfloat[][NeuMF]{
\includegraphics[width=0.45\linewidth]{Figures/eval/regret-movielens-1m-eta-0.5.pdf}
}
\hfil
\caption{Cumulative regret of Zeus vs. Grid Search across all workloads.}
\label{fig:appendix-regret}
\end{figure}
Figure~\ref{fig:appendix-regret} shows the cumulative regret of {Zeus\xspace} and Grid Search over job recurrences for all six workloads.
In general, {Zeus\xspace} converges to a better knob than Grid Search while being faster.
\section{Search Paths for All Workloads}
\label{sec:appendix-searchpath}
\begin{figure}[!htbp]
\centering
\hfil
\subfloat[][DeepSpeech2]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-zeus-librispeech-eta-0.5.pdf}
}
\hfil
\subfloat[][BERT (QA)]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-zeus-squad-eta-0.5.pdf}
}
\hfil
\subfloat[][BERT (SA)]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-zeus-sentiment140-eta-0.5.pdf}
}
\hfil
\subfloat[][ResNet-50]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-zeus-imagenet-eta-0.5.pdf}
}
\hfil
\subfloat[][ShuffleNet V2]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-zeus-cifar100-eta-0.5.pdf}
}
\hfil
\subfloat[][NeuMF]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-zeus-movielens-1m-eta-0.5.pdf}
}
\hfil
\caption{Search path of Zeus across all workloads.}
\label{fig:appendix-searchpath-zeus}
\end{figure}
\begin{figure}[!htbp]
\centering
\hfil
\subfloat[][DeepSpeech2]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-gs-librispeech-eta-0.5.pdf}
}
\hfil
\subfloat[][BERT (QA)]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-gs-squad-eta-0.5.pdf}
}
\hfil
\subfloat[][BERT (SA)]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-gs-sentiment140-eta-0.5.pdf}
}
\hfil
\subfloat[][ResNet-50]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-gs-imagenet-eta-0.5.pdf}
}
\hfil
\subfloat[][ShuffleNet V2]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-gs-cifar100-eta-0.5.pdf}
}
\hfil
\subfloat[][NeuMF]{
\includegraphics[width=0.45\linewidth]{Figures/eval/cov-heat-gs-movielens-1m-eta-0.5.pdf}
}
\hfil
\caption{Search path of Grid Search across all workloads.}
\label{fig:appendix-searchpath-gs}
\end{figure}
Figures~\ref{fig:appendix-searchpath-zeus} and~\ref{fig:appendix-searchpath-gs} respectively show the search path of Zeus and Grid Search in the 2D configuration space.
Thanks to the decoupling of batch size and power limit, {Zeus\xspace} is able to more efficiently navigate the search space and converge to a knob while consuming less energy and time during exploration.
\section{Additional Sensitivity Analysis}
\label{sec:appendix-sensitivity}
\begin{figure}[!t]
\centering
\subfloat[][ETA]{
\includegraphics[width=0.6\linewidth]{Figures/eval/sensitivity-eta-e-all.pdf}
}
\hfil
\subfloat[][TTA]{
\includegraphics[width=0.6\linewidth]{Figures/eval/sensitivity-eta-t-all.pdf}
}
\caption{Impact of priority knob $\eta$ on ETA and TTA.}
\label{fig:appendix-sensitivity-etaknob}
\end{figure}
Figure~\ref{fig:appendix-sensitivity-etaknob} compares both the energy consumption and training time for {Zeus\xspace} against Default.
We also calculate and plot the geometric mean across all jobs.
The result shows that with higher $\eta$, {Zeus\xspace} prioritizes reducing energy consumption over time, leading to higher improvement factor of energy, and vice versa.
\section{Performance of {Zeus\xspace} on All GPUs}\label{sec:appendix-trace-foi-gpus}
\begin{figure}[!t]
\centering
\subfloat[][Energy Consumption (V100)]{
\includegraphics[width=0.46\linewidth]{Figures/eval/cost-e-foi-v100-eta-0.5.pdf}
}
\hfil
\subfloat[][Training Time (V100)]{
\includegraphics[width=0.46\linewidth]{Figures/eval/cost-t-foi-v100-eta-0.5.pdf}
} \\
\subfloat[][Energy Consumption (A40)]{
\includegraphics[width=0.46\linewidth]{Figures/eval/cost-e-foi-a40-eta-0.5.pdf}
}
\hfil
\subfloat[][Training Time (A40)]{
\includegraphics[width=0.46\linewidth]{Figures/eval/cost-t-foi-a40-eta-0.5.pdf}
} \\
\subfloat[][Energy Consumption (RTX6000)]{
\includegraphics[width=0.46\linewidth]{Figures/eval/cost-e-foi-rtx6000-eta-0.5.pdf}
}
\hfil
\subfloat[][Training Time (RTX6000)]{
\includegraphics[width=0.46\linewidth]{Figures/eval/cost-t-foi-rtx6000-eta-0.5.pdf}
} \\
\subfloat[][Energy Consumption (P100)]{
\includegraphics[width=0.46\linewidth]{Figures/eval/cost-e-foi-p100-eta-0.5.pdf}
}
\hfil
\subfloat[][Training Time (P100)]{
\includegraphics[width=0.46\linewidth]{Figures/eval/cost-t-foi-p100-eta-0.5.pdf}
}
\caption{Energy and time consumption of DNN training, normalized against Default for DNN training. Results measured on (a) NVIDIA A40 GPU, (b) NVIDIA V100 GPU, (c) NVIDIA RTX6000 GPU and (d) NVIDIA P100 GPU.}
\label{fig:appendix-trace-foi-gpus}
\end{figure}
Figure~\ref{fig:appendix-trace-foi-gpus} presents the energy and time consumption of all workloads on four different generations NVIDIA GPUs: Ampere (A40), Volta (V100), Turing (RTX6000), and Pascal (P100).
The overall trends hold for all GPUs.
\section{{Zeus\xspace} Algorithm Design}
\label{sec:design}
In this section, we delve into the details of how {Zeus\xspace} selects the best batch size and GPU power limit to optimize the overall cost of recurrent DNN training tasks.
We first present the optimization problem formulation and how we decouple the optimizations of batch size and power limit (\S\ref{sec:design-components}).
Next, we show how to optimize power limit (\S\ref{sec:design-plo}) and batch size (\S\ref{sec:design-bso}) under the decoupled framework.
We conclude by discussing how we address common challenging scenarios (\S\ref{sec:extensions}).
\subsection{Problem Formulation}\label{sec:design-components}
The objective of {Zeus\xspace} is to minimize the cost of a recurring job by automatically exploring the feasible set of batch sizes $\mathcal{B}$ and power limits $\mathcal{P}$.
In essence, we neither want to incur too much cost searching for the optimal configuration, nor do we want to miss it.
Minimizing the \emph{cumulative} cost of the job over recurrences captures the implicit tradeoff between exploration and exploitation.
Put formally in terms of the cost function defined by Equation~\ref{eq:cost-metric}, our objective becomes
\begin{equation}
\begin{aligned}
\label{eq:optimization-problem}
\min_{b,p} \quad & \sum_{t=1}^T {C(b_t,p_t;\eta)} \\
\textrm{s.t.} \quad & b_t \in \mathcal{B}, p_t \in \mathcal{P}, \forall t \in [1, T],
\end{aligned}
\end{equation}
where $b_t$ and $p_t$ \revised{respectively} denote the batch size and power limit chosen at the $t$th recurrence of the job, \revised{and $b$ and $p$ are vectors of length $T$}
This is a challenging problem without modification, mainly because the size of the search space can be in the order of hundreds, and each value of $C(b, p; \eta)$ inside the search space can only be obtained by running DNN training until it reaches the target metric.
However, further expanding the cost function (Equation~\ref{eq:cost-tta-power}) allows us to \emph{decouple} the exploration of batch size and power limit, making the problem more tractable:
\begin{equation}
\label{eq:cost-breakdown-epoch}
\begin{aligned}
& C(b,p;\eta) \\
& = \left(\eta \cdot \mathtt{AvgPower}(b,p) + (1-\eta) \cdot \mathtt{MAXPOWER}\right) \cdot \mathtt{TTA} (b,p) \\
& = \mathtt{Epochs}(b) \cdot \frac{\eta \cdot \mathtt{AvgPower}(b,p) + (1-\eta) \cdot \mathtt{MAXPOWER}}{\mathtt{Throughput}(b,p)}.
\end{aligned}
\end{equation}
where $\mathtt{Epochs(b)}$ denotes the number of epochs needed to reach the target, and $\mathtt{Throughput(b,p)}$ epochs per second.
We find two key insights that allow the decoupling of batch size $b$ and power limit $p$:
\begin{denseenum}
\item Given $b$, $\mathtt{AvgPower}(b,p)$ and $\mathtt{Throughput}(b,p)$ can be profiled quickly during training for all possible choices of $p$.
This is due to the iterative nature of DNN training, yielding stable power and throughput estimations even with a small number of iterations
\item $\mathtt{Epochs}(b)$ is not affected by the choice of $p$ as changing the power limit does not change what is computed.
\end{denseenum}
This implies that the optimal power limit, given any batch size, can be determined independently based on online profiling.
Moreover, since any choice of batch size is automatically accompanied by the optimal power limit, our search space is reduced to the set of batch sizes $\mathcal{B}$.
Formally put, we have decoupled the problem in Equation~\ref{eq:optimization-problem} into an equivalent two-level optimization proble
\begin{equation}
\begin{aligned}
\label{eq:bso-optimization}
\min_{b \in \mathcal{B}\revised{^T}} \; & \sum_{t=1}^T {\mathtt{Epochs}(b_t) \cdot \mathtt{EpochCost}(b_t; \eta)} \\
\end{aligned}
\end{equation}
where
\begin{equation}
\begin{aligned}
\label{eq:plo-optimization}
& \mathtt{EpochCost}(b_t; \eta) \\
& = \min_{p_t \in \mathcal{P}} \; \frac{\eta \cdot \mathtt{AvgPower}(b_t,p_t) + (1-\eta) \cdot \mathtt{MAXPOWER}}{\mathtt{Throughput}(b_t,p_t)}. \\
\end{aligned}
\end{equation}
When a job arrives, {Zeus\xspace} will first decide which batch size to use based on Equation~\ref{eq:bso-optimization} (\S\ref{sec:design-bso}).
Then, based on the batch size, {Zeus\xspace} will pick the optimal power limit based on Equation~\ref{eq:plo-optimization} (\S\ref{sec:design-plo}).
\subsection{Optimizing the Power Limit}\label{sec:design-plo}
We start with how {Zeus\xspace} determines the optimal power limit based on Equation~\ref{eq:plo-optimization}, given a choice of the batch size.
As highlighted earlier, we leverage the iterative nature of DNN training and the recurrent nature of jobs in production DNN training workflows.
When a job with batch size decision $b$ is submitted, our just-in-time (JIT) profiler is triggered and checks if this batch size had been profiled before.
For \revised{an unseen batch size $b$}, it profiles $\mathtt{AvgPower}(b,p)$ and $\mathtt{Throughput}(b,p)$ for all possible power limits $p$ during the first epoch of the job by partitioning the epoch into slices at iteration boundaries and dynamically changing the GPU power limit for each slice.
The profile information is fed back to {Zeus\xspace}, and the optimal power limit of the batch size is determined by solving Equation~\ref{eq:plo-optimization}.
The rest of the epochs are executed with the optimal power limit.
Our \emph{online} JIT profiling approach consumes strictly less time and energy compared to offline profiling before running the job, because the profiling process itself contributes to training without affecting its accuracy.
\revised{We show that JIT profiling incurs negligible overhead in Section~\ref{sec:eval-overhead}.}
\subsection{Optimizing the Batch Size}\label{sec:design-bso}
Now we focus on how {Zeus\xspace} determines the batch size $b_t$ for each job recurrence $t$ that optimizes Equation~\ref{eq:bso-optimization}.
As seen in Section~\ref{sec:design-plo}, $\mathtt{EpochCost(b_t;\eta)}$ is a cheap and deterministic function that identifies the optimal power limit for any batch size $b_t$ and returns the optimal cost of one epoch.
Thus, we may limit our exploration to choosing the optimal batch size because whichever batch size we choose, the optimal power limit will accompany it.
Due to the unpredictable and stochastic nature of DNN training, picking out the optimal batch size without adequate exploration is difficult.
Hence, a good solution must
(1) incorporate such nature of DNN training into its exploration process, and (2) intelligently tradeoff the cost of exploring for potentially better batch sizes and the gain of exploiting batch sizes that are already known to be good.
\paragraph{Grid search is suboptimal.}
We argue that exhaustively going through all batch sizes and selecting the one with the smallest cost is still suboptimal due to the stochastic nature of DNN training.
That is, because the cost of a DNN training job can differ even when executed with the exact same configurations, it must be modeled as a \emph{cost distribution} with unknown mean and variance.
Although performing several trials for each batch size may yield a better estimation of the mean cost, such a strategy leads to \emph{high exploration cost} because it does not quickly rule out obviously suboptimal batch sizes.
\paragraph{Multi-Armed Bandit formulation.}
{Zeus\xspace} aims to explore the cost of different batch sizes and converge to the optimal batch size, while not incurring too much exploration cost.
{Zeus\xspace} formulates the problem as a Multi-Armed Bandit (MAB) with $T$ trials and $B$ arms, where each trial corresponds to a recurrence of the job and each arm to a batch size in $\mathcal{B}$.
MAB is a good fit to our problem scenario in that it \revised{captures the stochasticity of DNN training by modeling the cost of each batch size as a random variable.}
\revised{Specifically, we choose the} Gaussian distribution~\cite{ts-tutorial} due to its representational flexibility.
The objective of the MAB formulation is to minimize the \emph{cumulative cost regret} defined as
\begin{equation}
\label{eq:cumulative-cost-regret}
\sum_{t=1}^T { \mathtt{Regret}(b_t;\eta) }
\end{equation}
where the regret of choosing $b_t$ is defined as
\begin{equation}
\label{eq:cost-regret}
\begin{aligned}
& \mathtt{Regret}(b_t;\eta) \\
& = \mathtt{Epochs}(b_t) \cdot \mathtt{EpochCost}(b_t;\eta) - \min_{b,p} \; \mathtt{Cost}(b, p; \eta).
\end{aligned}
\end{equation}
Minimizing cumulative cost regret aligns with our objective in Equation~\ref{eq:bso-optimization}.
\paragraph{Thompson Sampling.}
\begin{algorithm}[t]
\DontPrintSemicolon
\SetNoFillComment
\let\oldnl\nl
\newcommand{\renewcommand{\nl}{\let\nl\oldnl}}{\renewcommand{\nl}{\let\nl\oldnl}}
\algrule
\KwIn{Batch sizes $\mathcal{B}$\newline
Belief posterior parameters $\hat{\mu}_b$ and $\hat{\sigma}_b^2$}
\KwOut{Batch size to run $b^*$
\SetKwFunction{Predict}{Predict}
\SetKwProg{Function}{Function}{:}{}
\algrule
\renewcommand{\nl}{\let\nl\oldnl} \Function{\Predict{$\mathcal{B}$, $\hat{\mu}_b$, $\hat{\sigma}_b^2$}}{
\ForEach{\normalfont{batch size} $b \in \mathcal{B}$}{
\tcc{Sample from the belief distribution}
Sample $\hat{\theta}_b \sim \mathcal{N}(\hat{\mu}_b, \hat{\sigma}_b^2)$\;
}
\tcc{Select the arm with smallest mean cost sample}
$b^* \gets \argmin_b \hat{\theta}_b$\;
}
\algrule
\caption{Gaussian Thompson Sampling: Choosing the next batch size to run (\texttt{Predict})}\label{algo:bso-predict}
\end{algorithm}
\begin{algorithm}[t]
\DontPrintSemicolon
\SetNoFillComment
\let\oldnl\nl
\newcommand{\renewcommand{\nl}{\let\nl\oldnl}}{\renewcommand{\nl}{\let\nl\oldnl}}
\algrule
\KwIn{Batch size $b$ and observed cost $C$\newline
Previous cost observations $C_b$ for $b$\newline
Belief prior parameters $\hat{\mu}_0$ and $\hat{\sigma}_0^2$}
\KwOut{Belief posterior parameters $\hat{\mu}_b$ and $\hat{\sigma}_b^2$}
\SetKwFunction{Observe}{Observe}
\SetKwProg{Function}{Function}{:}{}
\algrule
\renewcommand{\nl}{\let\nl\oldnl} \Function{\Observe{$b$, $C$, $\mathcal{C}_b$, $\hat{\mu}_0$, $\hat{\sigma}_0^2$}}{
\tcc{Add the most recent cost \revised{observation} to history}
$\mathcal{C}_b \gets \mathcal{C}_b \cup \{ C \}$
\tcc{Compute the variance of the cost}
$\tilde{\sigma}^2 \gets Var \left( \mathcal{C}_b \right)$\label{algo:bso-observe-cost-var}
\tcc{Compute the belief distribution's posterior variance}
$\hat{\sigma}_b^2 \gets {\left( \frac{1}{\hat{\sigma}_0^2} + \frac{|C_b|}{\tilde{\sigma}^2} \right)}^{-1}$
\tcc{Compute the belief distribution's posterior mean}
$\hat{\mu}_b \gets \hat{\sigma}_b^2 \left(\frac{\hat{\mu}_0}{\hat{\sigma}_0^2} + \frac{Sum(\mathcal{C}_b)}{\tilde{\sigma}^2} \right)$
}
\algrule
\caption{Gaussian Thompson Sampling: Updating the belief distribution (\texttt{Observe})}\label{algo:bso-observe}
\end{algorithm}
We adopt the Thompson Sampling~\cite{ts-tutorial} policy for the MAB formulation to tradeoff exploration and exploitation, not only because it is known to perform well in practice~\cite{empirical-ts,ts-tutorial} and had successful adoption recently~\cite{marcus2021bao, narya}, but also because its modeling assumptions fit our problem scenario well.
At a high level, Thompson Sampling is an online procedure that refines its \emph{belief} about the \emph{mean cost} of each arm (batch size) based on experience.
At each recurrence, the belief is used to pick the arm with the lowest estimated mean cost (Algorithm~\ref{algo:bso-predict}), and the belief is updated based on the actual cost observed (Algorithm~\ref{algo:bso-observe}).
Specifically, the cost distribution is modeled as a Gaussian distribution with unknown mean $\theta_b$.
Then, the belief about $\theta_b$ is modeled with its conjugate prior distribution, which is also a Gaussian distribution~\cite{conjugate}.
That is, $\theta_b \sim \mathcal{N}(\hat{\mu}_b,\hat{\sigma}_b^2)$.
Here it is important to note that $1/\hat{\sigma}_b^2$ can be thought as of how confident \revised{the policy} is in its belief about that arm, with the confidence increasing as it accumulates more observations of the cost of choosing that arm.
Then, Thompson Sampling automatically balances exploration and exploitation by choosing the arm with the smallest mean cost sample $\hat{\theta}_b \sim \mathcal{N}(\hat{\mu}_b, \hat{\sigma}_b^2)$ (Algorithm~\ref{algo:bso-predict}).
With low confidence (high variance), $\hat{\theta}_b$ will be dispersed across a wider range of costs, having higher chances of getting chosen even if some of its initial observations showed high cost.
In contrast, when the arms observed a lot of cost samples and the confidence is high (low variance), $\hat{\theta}_b$ is likely to be centered around the mean observed cost, allowing the exploitation of arms that are known to be good.
After the actual cost of an arm is observed, the belief parameters of that arm are updated using the Bayes Rule~\cite{ts-tutorial} (Algorithm~\ref{algo:bso-observe}).
The belief prior parameters $\hat{\mu}_0$ and $\hat{\sigma}_0^2$ reflect prior belief about the mean cost of using the batch size for training and the confidence of such belief.
Hence, the choice of prior parameters serve as a way to initialize the arms such that they reflect prior knowledge about the cost of each arm.
If such information is not available, which is our default assumption, it is also possible to initialize the arms with a flat prior that assumes no prior knowledge -- in our case, this is a Gaussian distribution with zero mean and infinite variance.
In contrast to grid search, our formulation using MAB and Thompson Sampling meets the two requirements mentioned earlier.
That is, MAB inherently incorporates the stochastic nature of DNN training in that it models cost as a \revised{random variable}.
Moreover, Thompson Sampling can quickly rule out batch sizes that are obviously suboptimal because the probability of a smaller mean cost being sampled from an arm that observed noticeably large cost is low.
\subsection{Extensions \revised{for Challenging Scenarios}}
\label{sec:extensions}
\paragraph{Handling unknown cost variance.}
Unlike conventional Gaussian Thompson Sampling applications, we may not assume that the variances of the cost of each arm are known.
That is, the cost variance (i.e., how much the cost will fluctuate even when training is run with the same batch size) is not known before any observation.
Moreover, the cost variance depends not only on the batch size, but also on the DNN's robustness to the randomness in parameter initialization and data loading, making it difficult to quantify at the time the MAB is constructed.
Hence, our approach is to \emph{learn} the cost variance as we observe cost samples (Line~\ref{algo:bso-observe-cost-var} in Algorithm~\ref{algo:bso-observe}).
\begin{figure}[!t]
\centering
\includegraphics[trim=220 220 320 175,clip,width=0.9\linewidth]{Figures/design/algo2stages.pdf}
\vspace*{-2mm}
\caption{An example of \revised{batch sizes chosen by {Zeus\xspace} for a recurring job.}
Each point is a recurrence. \revised{During pruning, {Zeus\xspace} explores each batch size 2 times in order to observe the cost variance (Line~\ref{algo:bso-observe-cost-var} in Algorithm~\ref{algo:bso-observe}).}}
\label{fig:design-algo-pruning}
\end{figure}
\paragraph{Handling stragglers during exploration.}
There may be cases where an exploratory job does not reach the target metric within a reasonable amount of \revised{cost}, especially during the earlier exploration stage.
To handle this, we employ \emph{early stopping} and \emph{pruning}.
The intuition is that if a batch size does not reach the target metric even after incurring an exceedingly large cost, it is highly unlikely to be the optimal one.
For early stopping, we define a cost threshold $\beta \cdot \min_t C_t$, meaning that when the cost of the current job is to exceed $\beta$ times the minimum cost observed so far, we stop the job and retry with another batch size.
Here $\beta$ is a parameter to account for the stochastic nature of DL training.
By default, we choose $\beta=2$, with which we should be able to tolerate variations of TTA between different runs of the same configuration, which is usually less than the 14\% \cite{dawnbench-tta}.
\begin{figure}[!t]
\centering
\includegraphics[width=0.65\linewidth]{Figures/model/bs-eta-librispeech.pdf}
\vspace*{-1mm}
\caption{ETA of each batch size for DeepSpeech2 trained on LibriSpeech. Plots for rest of the workloads are in the Appendix~\ref{sec:appendix-eta-bs-pl}.}
\label{fig:design-algo-bseta}
\end{figure}
\begin{algorithm}[t]
\DontPrintSemicolon
\SetNoFillComment
\let\oldnl\nl
\newcommand{\renewcommand{\nl}{\let\nl\oldnl}}{\renewcommand{\nl}{\let\nl\oldnl}}
\algrule
\KwIn{Set of batch sizes $\mathcal{B}$\newline
Default batch size $b_0$\newline
Belief prior parameters $\hat{\mu}_0$ and $\hat{\sigma}_0^2$}
\algrule
\tcc{\revised{Exploration With Pruning}}
Recurrence $t \gets 0$\;
\SetKwFor{RepeatN}{repeat}{times}{end}
\RepeatN{2}{
Explore $b_0$\;
Explore $b < b_0$ until convergence failure\;
Explore $b > b_0$ until convergence failure\;
$\mathcal{B} \gets \{b: b~\textnormal{converged}\}$\;
$b_0 \gets b$ with smallest cost observed\;
$t \gets t + |\mathcal{B}|$\;
}
\tcc{Thompson Sampling}
\While{$t \le T$}{
$b^* \gets Predict(\mathcal{B}, \hat{\mu}_b, \hat{\sigma}_b^2 \; \forall b \in \mathcal{B})$\;
Run job with batch size $b^*$ and add cost to $\mathcal{C}_b$\;
\tcc{Update our belief of the mean cost}
$\hat{\mu}_b, \hat{\sigma}_b^2 \gets Observe(b, \mathcal{C}_b, \hat{\mu}_0, \hat{\sigma}_0^2)$\;
$t \gets t + 1$\;
}
\algrule
\caption{Gaussian Thompson Sampling Batch Size Optimizer.}\label{algo:bso-overall}
\end{algorithm}
For pruning, as illustrated in Figure~\ref{fig:design-algo-pruning}, we begin with the default batch size provided by the user and first try smaller batch sizes until we meet the minimum batch size or a batch size that fails to reach the target metric before the early stopping threshold.
The same process is repeated for batch sizes larger than the default batch size.
Then, only the batch sizes that reached the target metric are kept in the batch size set we explore.
After performing an initial round of pruning, the default batch size is updated to be the one with the smallest cost observed, and we perform pruning once more starting from the new default batch size.
The intuition behind our batch size pruning approach is the convexity we observe in the BS-ETA curve around the optimal batch size (See Figure~\ref{fig:design-algo-bseta}).
Moreover, pruning allows {Zeus\xspace} to quickly rule out batch sizes that are noticeably suboptimal (typically too large, leading to more training epochs and loss of accuracy~\cite{largebatch1,largebatch2}, or too small, yielding gradients that are too noisy~\cite{smallbatch}), thus cutting down the cost of exploration.
The overall process is depicted in Algorithm~\ref{algo:bso-overall}.
\paragraph{Handling concurrent job submissions.}
Classic multi-armed bandit scenarios assume that the MAB immediately observes the cost of pulling an arm.
However, in a DNN training cluster, recurring jobs may overlap in their execution when a later job starts before the completion of an earlier job.
In this case, the MAB does not get to observe the cost of the earlier job at the time it has to decide the batch size for the later job.
For deterministic policies like~\cite{ucb1985,ucb2002}, this leads to duplication exploration of the same batch size back-to-back, reducing the efficiency of exploration.
However, Thompson Sampling naturally mitigates this problem without modification because deciding the next batch size to explore ($\mathtt{Predict}$) is a random function.
That is, because Thompson Sampling \emph{samples} the estimated mean cost from each arm's belief distribution and returns the arm with the lowest sampled value, concurrent jobs \revised{can run} different batch sizes even if there was no information gained between the invocations of $\mathtt{Predict}$.
This is especially the case during the early stage of Thompson Sampling when the arms' belief distributions have large variances (low confidence), losing little exploration efficiency.
During the short initial \revised{pruning} phase, we run concurrent job submissions with the best-known batch size at that time.
As the best batch size constantly updates throughout the exploration stage, this strategy fairly distributes the additional exploration opportunities from concurrent job submissions to batch sizes that are known to converge.
We evaluate {Zeus\xspace}'s efficacy on handling concurrent job submissions in Section~\ref{sec:eval-alibaba}.
\paragraph{Handling data drift.}
In production training clusters, the data on which the model is trained shifts, which is one of the reasons why re-training is triggered~\cite{learning-context-drift,liang2022metashift}.
The implication of drift in the perspective of the MAB is that the cost distribution of each arm is non-stationary.
Thompson Sampling allows a simple modification that allows us to handle non-stationary cost distributions.
Since older cost observations become less and less relevant, we only operate on a window of $N$ most recent cost observations~\cite{nonstationary-bandit}, and the belief distributions will not take old observations into account.
\revised{Unlike exponential decay, windowing also allows the cost variance of the most recent observations to be estimated directly.}
When old history entries are evicted, computing the new parameters of the arm is also cheap thanks to the conjugate prior property.
This way, {Zeus\xspace} transparently adapts to data drifts in an \emph{online} manner, as we show in Section~\ref{sec:eval-data-drift}.
\section{\revised{Discussion}}
\label{sec:discussion}
\paragraph{Choice of configuration knobs.}
In this paper, we pick the batch size and GPU power limit as the configuration knobs for {Zeus\xspace} to optimize.
We choose these two to strike a balance in the tradeoff between the granularity of control and the size of the search space.
For instance, one can set the frequency and voltage for individual components on the GPU for more fine-grained control and potentially higher energy efficiency, but this would result in prolonged exploration in the bigger search space.
In contrast, we choose the GPU power limit, which effectively controls both frequency and voltage via DVFS and reduces the search space.
On the DL job configuration side, we pick the batch size as the knob for a similar reason.
Changing the batch size has a broader impact on energy consumption of end-to-end DNN training, because it affects both the training time and the average power consumption during training.
In comparison, other candidate configuration knobs such as learning rate fall short because they only affect the training time.
\paragraph{Hyperparameter optimization.}
Hyperparameter optimization is an important workload, where many DL training jobs (trials) are submitted with specific hyperparameters chosen from a user-defined search space~\cite{hpo,asha,hyperband,fluid}.
If the users submit these trials with a specific batch size, they can specify the feasible batch size set $\mathcal{B}$ to only contain that single batch size.
In this case, {Zeus\xspace} can still reduce energy consumption by searching for the optimal GPU power limit.
\paragraph{Supporting distributed training.}
{Zeus\xspace} currently only supports single-node training, but it can easily be extended to support distributed scenarios.
Since the same type of GPU will have the same time and power consumption characteristics, we can apply the same power limit configuration across all GPUs to avoid stragglers.
The definition of cost can be extended to sum over the time and energy consumption of all GPUs participating in training, and all other components in our solution can remain identical.
\paragraph{Supporting heterogeneous GPUs.}
Our solution assumes that the training job runs on the same type of GPU across all of its recurrences.
However, in practice, this may not always be possible due to varying resource contention or availability.
It is straightforward to add support for heterogeneous GPUs under our formulation.
That is, cost values observed from one GPU can be \emph{translated} to values that represent the characteristics of another GPU.
As shown in Equation~\ref{eq:bso-optimization}, energy-time cost can be written as the product of $\mathtt{Epochs(b)}$ and $\mathtt{EpochCost}(b;\eta)$.
Here, the former term is independent with the choice of the GPU.
Moreover, the latter term can be quickly profiled on any GPU because it consists of only $\mathtt{AvgPower}(b,p)$ and $\mathtt{Throughput}(b,p)$.
Thus, we can obtain cost values that represent the new GPU by quickly profiling $\mathtt{EpochCost}(b;\eta)$ for each batch size on the new GPU and multiplying it with $\mathtt{Epochs}(b)$ observed from the previous GPU.
These translated cost observations can then be used to learn a new MAB that specializes on the new GPU.
\section{Evaluation}
\label{sec:eval}
\begin{table*}[]
\centering
\begin{tabular}{c c c c c c}
\hline
\textbf{Task} & \textbf{Dataset} & \textbf{Model} & \textbf{Optimizer} & \textbf {$\mathbf{b_0}$ } & \textbf{Target Metric} \\
\hline \hline
Speech Recognition & LibriSpeech~\cite{librispeech} & DeepSpeech2~\cite{deepspeech} & AdamW~\cite{adamw} & 192 & WER = 40.0\% \\
Question Answering & SQuAD~\cite{squad-paper} & BERT (QA)~\cite{bert} & AdamW~\cite{adamw} & 32 & F1 = 84.0 \\
Sentiment Analysis & Sentiment140~\cite{twitter-sentiment} & BERT (SA)~\cite{bert} & AdamW~\cite{adamw} & 128 & Acc. = 84\% \\
Image Classification & ImageNet~\cite{imagenet} & ResNet-50~\cite{resnet} & Adadelta~\cite{adadelta} & 256 & Acc. = 65\% \\
Image Classification & CIFAR-100~\cite{cifar-paper} & ShuffleNet-v2~\cite{shufflenetv2} & Adadelta~\cite{adadelta} & 1024 & Acc. = 60\% \\
Recommendation & MovieLens-1M~\cite{movielens} & NeuMF ~\cite{ncf} & Adam~\cite{adam} & 1024 & NDCG = 0.41 \\
\hline
\end{tabular}
\caption{Models and datasets used in our evaluation. The provided target metrics is the target for each training job. Here $b_0$ denotes the default batch size presented in the original work when feasible, otherwise we choose the maximum batch size which can consistently reach the target. The BERT(QA) and BERT(SA) means fine-tuning BERT on the tasks of question answering and sentiment analysis, respectively.}
\label{tbl:eval-workloads}
\end{table*}
We evaluate {Zeus\xspace}'s effectiveness in terms of navigating the energy-time tradeoff.
Our key findings are as follows:
\begin{denseenum}
\item {Zeus\xspace} reduces energy consumption by {15.3\%--75.8\%}.
It achieves this by trading off small performance for jobs that are already throughput-optimal; otherwise, it reduces training time by up to {60.1\%} too (\S\ref{sec:eval-performance}).
\item {Zeus\xspace} \revised{quickly converges to optimal configurations} (\S\ref{sec:eval-performance}).
\item {Zeus\xspace} can handle workloads with data drift (\S\ref{sec:eval-data-drift}) and overall incurs low overhead (\S\ref{sec:eval-overhead}).
\item {Zeus\xspace} scales to multi-GPU settings (\S\ref{sec:eval-multigpu}) and provides consistent savings across four generations of GPUs (\S\ref{sec:eval-sensitivity}).
\end{denseenum}
\subsection{Experimental Setup}
\label{sec:methodology}
\begin{table}[]
\centering
\smaller[2]
\begin{tabular}{l|ll|ll}
\hline
\textbf{Node} & \multicolumn{2}{l|}{\textbf{GPU Specification}} & \multicolumn{2}{l}{\textbf{Host Specification}} \\ \hline \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}HPE Apollo\\ 6500 Gen10 Plus\\ A40 $\times$ 4\end{tabular}} & \multicolumn{1}{l|}{Model} & A40 PCIe & \multicolumn{1}{l|}{CPU} & AMD EPYC 7513 \\
& \multicolumn{1}{l|}{VRAM} & 48GB & \multicolumn{1}{l|}{RAM} & 512GB DDR4-3200 \\
& \multicolumn{1}{l|}{mArch.} & Ampere & \multicolumn{1}{l|}{Disk} & 960GB NVMe SSD \\ \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}CloudLab~\cite{cloudlab}\\ r7525\\ V100 $\times$ 2\end{tabular}} & \multicolumn{1}{l|}{Model} & V100 PCIe & \multicolumn{1}{l|}{CPU} & AMD EPYC 7542 \\
& \multicolumn{1}{l|}{VRAM} & 32GB & \multicolumn{1}{l|}{RAM} & 512GB DDR4-3200 \\
& \multicolumn{1}{l|}{mArch.} & Volta & \multicolumn{1}{l|}{Disk} & 2TB 7200rpm HDD \\ \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}Chameleon\\ Cloud~\cite{chameleoncloud}\\ RTX6000\end{tabular}} & \multicolumn{1}{l|}{Model} & RTX6000 & \multicolumn{1}{l|}{CPU} & Xeon Gold 6126 \\
& \multicolumn{1}{l|}{VRAM} & 24GB & \multicolumn{1}{l|}{RAM} & 192GB \\
& \multicolumn{1}{l|}{mArch.} & Turing & \multicolumn{1}{l|}{Disk} & 256GB SSD \\ \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}Chameleon\\ Cloud~\cite{chameleoncloud}\\ P100 $\times$ 2\end{tabular}} & \multicolumn{1}{l|}{Model} & P100 & \multicolumn{1}{l|}{CPU} & Xeon E5-2670 v3 \\
& \multicolumn{1}{l|}{VRAM} & 16GB & \multicolumn{1}{l|}{RAM} & 128GB \\
& \multicolumn{1}{l|}{mArch.} & Pascal & \multicolumn{1}{l|}{Disk} & 1TB HDD \\
\hline
\end{tabular}
\caption{\revised{Hardware used in the evaluation.}}
\label{tbl:eval-gpus}
\end{table}
\paragraph{Testbed Setup.}
We evaluate {Zeus\xspace} with four generations of NVIDIA GPUs \revised{as specified in Table~\ref{tbl:eval-gpus}}.
\paragraph{Workloads.}\label{sec:eval-workloads.}
Table~\ref{tbl:eval-workloads} summarizes our workloads.
The default batch size ($b_0$) is chosen from the original model publication when available; otherwise, it is set to be the maximum batch size which consistently achieves the target accuracy.
\revised{In terms of learning rate, models trained with the Adadelta~\cite{adadelta} optimizer do not require an initial learning rate.}
\revised{For optimizers that do require an initial learning rate, we made our best effort in choosing a batch size and learning rate pair that achieves reasonable accuracies by experimenting with values from the original publication of the model and those discovered by popular DL frameworks~\cite{huggingface}.}
\revised{After collecting the initial batch size and learning rate pairs, when we scale the batch size, we applied Square Root Scaling~\cite{lr-scaling-sqr} for adaptive optimizers such as Adam~\cite{adam} following recent theoretical results~\cite{lr-scaling-prescription}.}
\paragraph{Baselines.}
We compare against the following baselines:
\begin{denseenum}
\item \emph{Default ($b=b_0$, $p=\mathtt{MAXPOWER}$).} This is often the default configuration used by practitioners, where the GPU power limit is set to, or rather \emph{not changed from}, the maximum.
This is the most conservative baseline with no exploration.
\item \emph{Grid Search with Pruning.} This one tries out one configuration of $(b, p)$ for each recurrence of the job and selects the best one.
We optimize na\"ive grid search by having it prune out batch sizes that failed to reach the target metric.
\end{denseenum}
\paragraph{Metric.}
Our primary metrics are ETA (energy consumption) and TTA (training time).
Ideally, we want to reduce both; but due to their tradeoff, sometimes it may not be possible to simultaneously do both.
\paragraph{Defaults.}
All experiments are done on NVIDIA V100 GPUs, unless otherwise mentioned.
By default, we highlight $\eta=0.5$ to strike a balance between ETA and TTA.
Later, we sweep $\eta$ from 0 to 1 (\S\ref{sec:eval-sensitivity}).
The early-stopping threshold $\beta$ is set to 2, and we also sweep $\beta$ from 1.5 to 5 (\S\ref{sec:eval-sensitivity}).
\paragraph{\revised{Methodology.}}
\revised{Due to resource constraints and environmental concerns, we cannot afford to repeatedly train all of our workloads with various configurations end-to-end hundreds of times sequentially.}
\revised{However, similar to how {Zeus\xspace} \emph{decouples} the exploration of batch size and power limit, we may apply the same decoupling in our experimentation.}
\revised{That is, we instead take a trace-driven approach, where we collect two kinds of trace data:}
\begin{denseenum}
\item \revised{Training trace. We train all possible combinations of models and batch sizes until convergence and record the number of epochs the model took to reach its target accuracy. We repeat this with four different random seeds for every combination to capture the stochasticity in DNN training.}
\item \revised{Power trace. We use our JIT profiler to collect the throughput and average power consumption of all possible combinations of model, batch size, and power limit.}
\end{denseenum}
\revised{We then replay these traces when we need to train a model and reconstruct its TTA and ETA values in order to evaluate the decisions made by {Zeus\xspace} and baselines.}
\revised{Moreover, since we have access to all the possible choices and their outcomes, we also know the optimal choice.}
\revised{Therefore, with the traces, we can evaluate the regret achieved by {Zeus\xspace} and baselines.}
\revised{Note that {Zeus\xspace} does not directly learn from these traces (which would be offline-profiling), but instead only learns from the \emph{replay} of these traces in an online fashion.}
\revised{While the aforementioned trace-driven method is used widely throughout our evaluation, we run {Zeus\xspace} end-to-end for the evaluation of handling data drift (\S\ref{sec:eval-data-drift}) because it is more expensive to construct the trace for the drifting dataset.}
\subsection{{Zeus\xspace} Performance}
\label{sec:eval-performance}
In this section, we evaluate the performance of {Zeus\xspace} in terms of energy consumption and training time as well as the convergence characteristics of our Multi-Armed Bandit algorithm.
Each experiment is run across multiple recurrences of DNN training jobs.
We select the recurrence number to be $2 \cdot |\mathcal{B}| \cdot |\mathcal{P}|$, so that the Grid Search baseline finishes exploration and also has plenty of chances to exploit its choice.
\begin{figure}[!t]
\centering
\subfloat[][Energy Consumption]{
\includegraphics[width=0.86\linewidth]{Figures/eval/cost-e-foi-v100-eta-0.5.pdf}
\label{fig:eval-cost-foi-energy}
} \\
\subfloat[][Training Time]{
\includegraphics[trim=0 0 0 22,clip,width=0.86\linewidth]{Figures/eval/cost-t-foi-v100-eta-0.5.pdf}
\label{fig:eval-cost-foi-time}
}
\caption{{Zeus\xspace} reduces energy consumption for all workloads. (a) energy consumption, (b) training time of each workload, normalized by the Default baseline. Results are computed with the last \revised{five} recurrences, capturing the knobs each method converged to.}
\label{fig:eval-cost-foi-v100}
\end{figure}
\paragraph{Improvements in ETA.}
Figure~\ref{fig:eval-cost-foi-energy} shows the energy consumption (ETA) of the last \revised{five} recurrences of {Zeus\xspace} and Grid Search w.r.t. the Default baseline, aiming to compare the final point each approach converged to.
{Zeus\xspace} reduces energy consumption (ETA) by up to {15.3\%--75.8\%} w.r.t. the baseline.
This is also comparable to the reduction we found by exhaustively searching through all the configurations in Section~\ref{sec:motivation} as well as by using Grid Search.
\paragraph{Tradeoff with TTA.}
Figure~\ref{fig:eval-cost-foi-time} shows the time consumption (TTA) of the last \revised{five} recurrences of {Zeus\xspace} and Grid Search w.r.t. the Default baseline.
Even though {Zeus\xspace} reduces training time by up to {60.1\%}, for some workloads TTA is increased by {12.8\%} (Figure~\ref{fig:eval-cost-foi-time}).
This is due to the tradeoff between ETA and TTA, which is the central focus of this paper.
This is especially true for workloads with \revised{a $b_0$ tuned for minimizing training time}, where there is little room for TTA improvement.
\begin{figure}[!t]
\centering
\hfil
\subfloat[][DeepSpeech2]{
\includegraphics[width=0.45\linewidth]{Figures/eval/regret-librispeech-eta-0.5.pdf}
}
\subfloat[][ResNet-50]{
\includegraphics[width=0.45\linewidth]{Figures/eval/regret-imagenet-eta-0.5.pdf}
}
\hfil
\vspace*{-2mm}
\caption{Cumulative regret of Zeus vs. Grid Search for (a) DeepSpeech2 and (b) ResNet-50. }
\label{fig:eval-regret}
\end{figure}
\paragraph{Cumulative regret.}
While {Zeus\xspace} and Grid Search perform close to each other, {Zeus\xspace} uses significantly smaller amount of resources to converge.
As a bandit-based solution, the effectiveness of our algorithm can be quantified via regret, the difference between the decision selected and the optimal choice (Equation~\ref{eq:cost-regret} in Section~\ref{sec:design-bso}).
Figure~\ref{fig:eval-regret} shows the cumulative regret of {Zeus\xspace} and Grid Search for DeepSpeech2 and ResNet-50.
The optimal configuration is \revised{identified separately} by an exhaustive parameter sweep.
We observe that in both workloads, {Zeus\xspace} is able to achieve better regret from the first job recurrence.
{Zeus\xspace} reaches the plateau in the cumulative regret earlier than Grid Search, which means it converges to the optimal solution earlier.
We observe similar results for other workload as well (Appendix~\ref{sec:appendix-regret}).
In the worst case, Grid Search \revised{results in $72\times$ more cumulative regret than {Zeus\xspace} until convergence}.
\paragraph{Convergence to a Pareto-optimal configuration.}
\begin{figure}[!t]
\centering
\subfloat[][{Zeus\xspace}]{
\includegraphics[width=0.75\linewidth]{Figures/eval/cov-heat-zeus-librispeech-eta-0.5.pdf}
}
\subfloat[][Grid Search]{
\includegraphics[width=0.75\linewidth]{Figures/eval/cov-heat-gs-librispeech-eta-0.5.pdf}
\label{fig:eval-covXY-gs}
}
\caption{Search paths of (a) {Zeus\xspace} and (b) Grid Search for Deepspeech2. The heatmap in the background shows the regret of each (Batch Size, Power Limit) configuration. Darker background denotes lower regret and therefore better configuration. The colored line with shifting color shows the search path, with darker color being later recurrences.}
\label{fig:eval-covXY-all}
\end{figure}
Despite having no information about the application beforehand, {Zeus\xspace} learns the energy characteristics of it online in a few iterations.
Figure~\ref{fig:eval-covXY-all} shows the search path of {Zeus\xspace} and Grid Search during training DeepSpeech2.
Due to the decoupling in \revised{the} optimization of power limit and batch size, {Zeus\xspace} explores the configuration space more efficiently and converges to the optimal configuration much faster.
We observe similar results for other workloads (see Appendix~\ref{sec:appendix-searchpath}).
\revised{Moreover}, in Figure~\ref{fig:eval-covXY-gs} we observe that Grid Search may not even converge to optimal configuration.
This is due to the stochastic nature of DNN training, with even the same batch size yielding different energy and time consumptions.
Hence, Grid Search may choose a suboptimal configuration when a suboptimal configuration luckily yields good energy and time consumptions.
\subsection{Trace-Driven Simulation Using the Alibaba Trace}
\label{sec:eval-alibaba}
Here we evaluate how {Zeus\xspace} can save energy and time consumption for DNN training in large clusters.
We run trace-driven simulation using the Alibaba \revised{GPU cluster} trace~\cite{alibaba-trace} which contains over 1.2 million jobs spanning a period of two months.
The Alibaba GPU cluster trace \revised{is suitable for our evaluation for} two reasons.
First, the trace identifies \revised{groups of} recurring jobs, and each job is annotated with its group ID.
Second, jobs \revised{within} the same group show overlap in their execution, allowing us to evaluate {Zeus\xspace}'s capability of handling concurrent job submissions with Thompson Sampling.
\begin{figure}[!t]
\centering
\subfloat[][Energy Consumption]{
\includegraphics[width=0.86\linewidth]{Figures/eval/alibaba-energy-v100-eta-0.5.pdf}
\label{fig:eval-alibaba-energy}
}
\hfil
\subfloat[][Training Time]{
\includegraphics[trim=0 0 0 22,clip,width=0.86\linewidth]{Figures/eval/alibaba-time-v100-eta-0.5.pdf}
\label{fig:eval-alibaba-time}
}
\caption{{Zeus\xspace} reduces energy consumption for all jobs in the Alibaba cluster trace~\cite{alibaba-trace}, compared to Grid Search and Default. (a) Energy consumption with {Zeus\xspace} comparing against baselines, (b) Training time of each type of workload. Both are normalized by the Default baseline. }
\label{fig:fig:alibaba-all}
\end{figure}
In order to assign job groups to the workload (Table~\ref{tbl:eval-workloads}) that best resembles its runtime, we remove jobs that did not successfully terminate and run K-Means clustering~\cite{kmeans} on the mean job runtime of each group to form six clusters.
Then, we match the six clusters with our six workloads in the order of their mean runtime.
When running simulation, in order to capture the intra-cluster runtime variation of each job, we scale the job runtime with the ratio of the job's original runtime to its cluster's mean runtime.
We compare {Zeus\xspace} with Default and Grid Search and plot the results in Figure~\ref{fig:fig:alibaba-all}.
Figure~\ref{fig:eval-alibaba-energy} shows the cumulative energy consumption of training using all three approaches.
{Zeus\xspace} outperforms both baselines for workloads of all types and sizes.
Note that there are scenarios where the Grid Search performs worse than Default, due to it wasting too much energy and time during the exploration stage.
Thanks to {Zeus\xspace}'s \emph{early stopping} and quick online power optimization, its energy and time cost during the exploration stage is significantly reduced.
Across all the models, {Zeus\xspace} reduces training energy usage by 7\%--52\%.
Figure~\ref{fig:eval-alibaba-time} shows the training time using {Zeus\xspace} to be increased by at most 16\%, and in many cases even decreased by up to 33\%.
Finally, similar to earlier experiments, {Zeus\xspace} had significantly lower cumulative regret than Grid Search.
\subsection{Handling Data Drift}\label{sec:eval-data-drift}
While there are previous works that attempt to identify and address data drift in general ML settings~\cite{learning-context-drift}, existing datasets are classification tasks based on small feature vectors~\cite{electricity-dataset,airlines-dataset}, completely synthetic~\cite{circle-dataset,hyperplane-dataset}, or \revised{private}~\cite{matchmaker}.
Therefore, we create and open-source a new sentiment analysis dataset called \emph{Capriccio\xspace} that is suitable for evaluating DNN models.
{Capriccio\xspace} consists of 1.6 million tweets over three months from the Sentiment140~\cite{twitter-sentiment} dataset, labeled with sentiment scores and the timestamp of the tweet.
We emulate data drift by capturing a sliding window of 500,000 tweets (roughly the amount of tweets in one month) at a time and moving the window forward by each day, generating 38 slices.
We skip empty dates to avoid having identical slices.
We train BERT~\cite{bert} on {Capriccio\xspace} with {Zeus\xspace} configured with a window size of 10, roughly corresponding to a time frame of two weeks on Twitter.
We plot the selected batch size for each recurrence (slice) and its corresponding ETA and TTA of training in Figure~\ref{fig:eval-drift-twitter}.
\revised{It can be seen that spikes in ETA and TTA (signaling that the current batch size may no longer be optimal) trigger the exploration of a batch size that is different from the one previously converged to.}
\begin{figure}[!t]
\centering
\includegraphics[width=0.8\linewidth]{Figures/eval/drift-cost-bs-with-tta.pdf}
\vspace*{-2mm}
\caption{\revised{Energy and time consumption of training BERT with {Zeus\xspace} on {Capriccio\xspace} and the batch size chosen for each slice.}}
\label{fig:eval-drift-twitter}
\end{figure}
\subsection{Overhead of JIT Profiling}\label{sec:eval-overhead}
Measurements with the Deepspeech2 model using the default batch size $b_0$ show that JIT profiling results in a 0.01\% increase in energy consumption and a 0.03\% increase in time consumption.
Such a tiny overhead is possible because the time needed to profile all power limits is very small (less than one minute) while one epoch of training spans hours (which is typical for DL workloads).
Measurements on ShuffleNet-v2, which has much shorter epoch duration, show that JIT profiling results in a 0.6\% increase in terms of time consumption and a 2.8\% reduction in energy consumption.
\subsection{Scaling to Multi-GPU}\label{sec:eval-multigpu}
While the primary focus of this paper is on single-GPU settings, in this section, we show that {Zeus\xspace} can be extended to \revised{single-node} multi-GPU training settings by profiling the power consumption of all GPUs that participate in training.
Extensions to distributed multi-GPU \revised{setups that involve network communication} is a potential future work.
Extending to multi-GPU allows us to compare our energy and time consumption with Pollux~\cite{pollux}, a state-of-the-art distributed cluster scheduler that dynamically tunes the batch size \emph{during} DNN training in order to maximize \emph{goodput}.
Training DeepSpeech2 on LibriSpeech on four NVIDIA A40 GPUs, {Zeus\xspace} consumes 12\% more time but 21\% less energy, comparing favorably.
We especially note that while Pollux does not take energy into account, {Zeus\xspace} allows the user to select a different \revised{energy-time} tradeoff point (e.g., speed up training but consume more energy) by selecting an appropriate $\eta$.
\subsection{Sensitivity Analysis and Ablation Studies}
\label{sec:eval-sensitivity}
\paragraph{Impact of $\eta$.}
\begin{figure}[!t]
\centering
\includegraphics[width=0.8\linewidth]{Figures/eval/eval-pareto-etaknob.pdf}
\vspace*{-2mm}
\caption{Pareto Front of DeepSpeech2 and how $\eta$ navigates it.}
\label{fig:eval-sensitivity-etaknob}
\end{figure}
To characterize the impact of $\eta$ as defined in Equation~\ref{eq:cost-metric}, we perform a sweep of $0 \leq \eta \leq 1$ when training DeepSpeech2 and plot the resulting optimal (TTA, ETA) in Figure~\ref{fig:eval-sensitivity-etaknob}.
We also plot the corresponding Pareto Front for reference.
We observe that the resulting (TTA, ETA) data points fall closely to the Pareto Front.
Moreover, we plot the lines along which the $C$ in Equation~\ref{eq:cost-metric} is a constant, shown as the dotted lines.
As expected, these lines form an envelope around the Pareto Front.
Additional sensitivity analysis for $\eta$ can be found in Appendix~\ref{sec:appendix-sensitivity}.
\paragraph{Impact of early-stopping threshold $\beta$.}
\begin{figure}[!t]
\centering
\includegraphics[width=0.65\linewidth]{Figures/eval/sensitivity-beta-all.pdf}
\vspace*{-2mm}
\caption{Relative cumulative energy consumption of {Zeus\xspace} across all jobs, w.r.t. the early-stopping threshold $\beta$.}
\label{fig:eval-sensitivity-beta}
\end{figure}
To study impact of the early-stopping threshold $\beta$, we sweep $\beta$ from 1.5 to 5 and measure the cumulative ETA across all jobs.
We calculate the difference in ETA relative to our default choice of $\beta=2.0$, and plot the result of all jobs as well as a geometric mean across all jobs in~Figure~\ref{fig:eval-sensitivity-beta}.
The result shows that the default $\beta=2.0$ chosen by {Zeus\xspace} achieves the lowest geometric mean across all jobs.
The intuition behind this is \revised{that when} $\beta$ is too low, {Zeus\xspace} prematurely stops exploratory runs, reducing the effectiveness of exploration.
In contrast, when $\beta$ is too high, it dilutes the benefit of early stopping which leads to inflated \revised{exploration cost}.
\paragraph{Impact of individual components.}
In order to show the gains from each component, we show the degradation of removing one component from {Zeus\xspace}: no early stopping (setting $\beta$ to infinity), no pruning (keeping a batch size that didn't reach the target accuracy), and no JIT profiling (profiling each power limit in different recurrences).
Figure~\ref{fig:eval-breakdown-foi-v100} shows the slowdown relative to {Zeus\xspace} after disabling these components.
We observe that the {Zeus\xspace} benefits mostly from early stopping.
\paragraph{Impact of GPU models.}
Figure~\ref{fig:eval-sensitivity-gpu} shows the geometric mean of ETA normalized against Default across all jobs.
\revised{{Zeus\xspace} achieves consistent ETA reductions} across four generations of NVIDIA GPUs.
\revised{See Appendix~\ref{sec:appendix-trace-foi-gpus} for all results.}
\begin{figure}[!t]
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[trim=0 6 0 0,clip,width=0.98\linewidth]{Figures/eval/breakdown-gmean-v100-eta-0.5.pdf}
\caption{Performance breakdown of {Zeus\xspace}.}
\label{fig:eval-breakdown-foi-v100}
\end{minipage}
\hfil
\begin{minipage}[b]{0.46\linewidth}
\centering
\includegraphics[width=0.98\linewidth]{Figures/eval/sensitivity-gpu-foi-gmean.pdf}
\caption{\revised{Normalized ETA w.r.t. GPU models.}}
\label{fig:eval-sensitivity-gpu}
\end{minipage}
\end{figure}
\section{Zeus\xspace Implementation}
\label{sec:impl}
{Zeus\xspace} is implemented as a Python library that can be imported into DNN training scripts.
\revised{The \texttt{ZeusDataLoader} class integrates with PyTorch~\cite{pytorch}.}
The class profiles power and throughput online by slicing epochs in iteration boundaries and invoking the NVML~\cite{nvml} library for power limit configuration and profiling.
We have observed that five seconds of profiling for each power limit is enough to yield stable results.
With the information, the optimal power limit can be automatically determined and applied.
Moreover, \texttt{ZeusDataLoader} monitors the cost incurred by training and early stops the job if needed.
Listing~\ref{lst:design-zeus-integration} shows an example training loop integrated with {Zeus\xspace}.
\begin{lstlisting}[float, language=Python, label=lst:design-zeus-integration, caption={Zeus\xspace} Integration Example]
from zeus import ZeusDataLoader
train_loader = ZeusDataLoader(
train_set, batch_size, max_epochs, target_metric)
eval_loader = ZeusDataLoader(eval_set, batch_size)
for epoch in train_loader.epochs(): # may early stop
for batch in train_loader:
# Learn from batch
for batch in eval_loader:
# Evaluate on batch
train_loader.report_metric(validation_metric)
\end{lstlisting}
\paragraph{Observer Mode.}
\texttt{ZeusDataLoader} supports \emph{Observer Mode}, where it profiles the power consumption and throughput of each power limit and determines the optimal one, but keeps the power limit at the maximum.
By doing so, without affecting time or energy consumption, \texttt{ZeusDataLoader} reports how much time and energy the job \emph{would have} consumed if the power limit were the optimal one, allowing the user to get an idea of the impact of using {Zeus\xspace}.
We believe that such a feature can encourage Zeus\xspace's adoption by informing users of its potential savings.
\section{Introduction}
\label{sec:intro}
\revised{Deep neural networks (DNNs) have received ubiquitous adoption in recent years across many data-driven application domains such as computer vision~\cite{imagenet,resnet,shufflenetv2}, natural language processing~\cite{bert,albert}, personalized recommendation~\cite{dnn-recommendation-fb,ncf}, and speech recognition~\cite{deepspeech}.}
\revised{To effectively support such growth, DNN models are predominantly trained in clusters of highly parallel and increasingly more powerful GPUs~\cite{dnn-hardware-survey, rnn-hardware-survey}.}
However, growing demand for computation ultimately translates to greater energy demand.
For instance, training the GPT-3 model~\cite{gpt3} consumes 1,287 megawatt-hour (MWh)~\cite{patterson2021carbon}, which is equivalent to 120 years of electricity consumption for an average U.S. household~\cite{us-household}.
\revised{This trend continues to grow: Meta reports an increasing electricity demand for AI, despite a 28.5\% operational power footprint reduction~\cite{fb-sustainable-ai}.}
Yet, existing literature on DNN training mostly ignores energy efficiency~\cite{greenai}.
We observe that \emph{common performance optimization practices for DNN training can lead to inefficient energy usage}.
\revised{For example, many recent works prescribe large \emph{batch sizes} for higher training throughput~\cite{imagenet1hour,incbs}.
However, we show that maximizing raw throughput may come at the cost of lower energy efficiency.}
Similarly, modern GPUs \revised{allow the configuration of a \emph{power limit} that caps its maximum power draw}, but existing solutions often ignore it.
Our analysis of four generations of NVIDIA GPUs shows that none of them are entirely power proportional, and drawing maximum power \revised{gives diminishing return}.
Indeed, carefully choosing the right batch size and GPU power limit can reduce energy consumption by {23.8\%--74.7\%} for diverse workloads (\S\ref{sec:motivation-eta-gap}).
Unfortunately, reducing energy consumption is not entirely free -- we discover that there is a tradeoff between energy consumption and training time for a given target accuracy (\S\ref{sec:motivation-pareto}).
Our characterization of the \revised{energy-time} Pareto frontier highlights two notable phenomena.
First, for a given training job, all Pareto-optimal configurations provide varying amounts of energy reductions in comparison to blindly using the maximum batch size and GPU power limit.
\revised{Second, the amount of reduction in energy consumption often has a non-linear relationship with the increase of training time.}
This raises a simple question: \emph{how do we automatically identify and navigate the tradeoff between energy consumption and training time for DNN training?}
In this paper, we present {Zeus\xspace} to address this question.
{Zeus\xspace} is a plug-in optimization framework that automatically configures the batch size and GPU power limit to minimize the overall energy consumption and training time for DNN training jobs (\S\ref{sec:overview}).
Unlike some recent works that only consider GPU-specific configurations \cite{dvfs-impact,dynamic-underclock-gpu}, {Zeus\xspace} simultaneously considers job- and GPU-related configurations.
\revised{Moreover}, it does not require per-job offline profiling or prediction model training~\cite{gpoeo,odpp}, both of which can be prohibitive in large clusters with heterogeneous hardware and time-varying workloads~\cite{alibaba-trace}.
Instead, {Zeus\xspace} \revised{takes an online exploration-exploitation approach} tailored to the characteristics of DNN training workflows.
\revised{That is, as new data flow into the pipeline, models need to be periodically re-trained~\cite{applied-ml-at-fb}, manifesting itself as \emph{recurring jobs} in production clusters~\cite{applied-ml-at-fb,alibaba-trace}.}
Leveraging this fact, {Zeus\xspace} automatically explores various configurations, measures corresponding gains or losses, and continuously adjusts its actions based on its measurements (\S\ref{sec:design}).
Designing such a solution is challenging due to two sources of uncertainty in DNN training.
First, due to the randomness introduced from DNN parameter initialization and data loading, the energy consumed until a DNN reaches its target accuracy varies even when training is run with the exact same configuration~\cite{randomness-DNN,dawnbench-tta}.
Thus, evaluating a configuration only once does not provide sufficient information about its \emph{expected} energy consumption.
Second, since both DNN models and GPUs have diverse architectures and unique energy characteristics~\cite{benchmark-ai-accelerators}, offline profiling results do not easily generalize to other DNNs and GPUs.
Aggravating these challenges is the large size of the possible configuration space, with each configuration taking hours or even days to evaluate.
{Zeus\xspace} can efficiently determine the optimal set of knobs in the configuration space by \emph{decoupling} the optimization of batch size and power limit without losing optimality.
Specifically, it captures the stochastic nature of DNN training by formulating the batch size optimization problem as a Multi-Armed Bandit (MAB) and runs online optimization under random observations using the Thompson Sampling policy~\cite{thompson1933likelihood}.
Additionally, {Zeus\xspace}'s just-in-time (JIT) energy profiler finds the optimal power limit while training is running, making {Zeus\xspace} a completely online optimization framework.
We have implemented {Zeus\xspace} and integrated it with PyTorch~\cite{pytorch} (\S\ref{sec:impl}).
Evaluation on a diverse workload consisting of speech recognition, image classification, NLP, and recommendation tasks shows that {Zeus\xspace} reduces energy consumption by \revised{{15.3\%--75.8\%}} and training time by {60.6\%} w.r.t. simply selecting the maximum batch size and maximum GPU power limit.
{Zeus\xspace} converges to optimal configuration among available ones quickly and can adapt to data drift effectively.
{Zeus\xspace}'s benefits expand to multi-GPU settings as well (\S\ref{sec:eval}).
In summary, we make the following contributions:
\begin{denseitemize}
\item To the best of our knowledge, we are the first to characterize the energy consumption vs. performance tradeoff for DNN training in terms of job- and GPU-specific configuration parameters.
\item We present an online optimization framework that can learn from and adapt to workload dynamics over time.
\item We implement and evaluate the optimizer in {Zeus\xspace} that integrates with existing DNN training workflows with little code change and negligible overhead, while enabling large benefits.
\end{denseitemize}
\revised{{Zeus\xspace} is open-source and available on GitHub.\footnote{\url{https://github.com/SymbioticLab/Zeus}}}
\section{Motivation}
\label{sec:motivation}
In this section, we present an overview of energy consumption characteristics of DNN training on GPUs, opportunities for reducing energy consumption, and conclude with characterizing the tradeoff between reducing energy consumption and improving training performance.
\subsection{DNN Training}
Modern DNNs are trained by going over a large dataset multiple times, where each pass over the dataset is termed an \emph{epoch}~\cite{goodfellowdl}.
One epoch of training consists of thousands of \emph{iterations} of gradient descent over equally sized mini-batches, with the \emph{batch size} affecting model accuracy,\footnote{In this paper, we specifically \revised{consider the} \emph{validation accuracy} of the model, which captures how well the model \revised{generalizes to} unseen data.} training throughput, and energy consumption.
The performance of DNN training is often measured in terms of time-to-accuracy (TTA) for a given target accuracy~\cite{dawnbench-tta}, and increasing training throughput (or precisely goodput~\cite{pollux}) leads to lower TTA.
\revised{Modern DNNs are predominantly trained on increasingly more powerful GPUs}, consuming more energy in the process~\cite{patterson2021carbon,fb-sustainable-ai,treehouse}.
Recent benchmarks show that GPUs are responsible for around 70\% of the total energy consumption during DNN training~\cite{gpu-70percent,measuring-carbon}.
\revised{In production GPU clusters, as new data flow into the machine learning pipeline, DNNs need to be periodically re-trained at intervals as short as every hour~\cite{applied-ml-at-fb}.}
\revised{This need manifests itself as \emph{recurring jobs} in the GPU cluster~\cite{applied-ml-at-fb,alibaba-trace}.}
\subsection{Opportunities for Improving Energy Efficiency}
\label{sec:motivation-eta-gap}
We \revised{highlight} two job and hardware configurations that can cause sizable energy inefficiency in DNN training:
(1) \revised{batch size}
and
(2) power limit of the GPU.
\paragraph{Impact of batch size on energy efficiency.}
The size of each mini-batch during DNN training (batch size) determines how many samples are processed in one iteration.
The higher it is, the faster we can go over the entire input dataset.
\revised{We observe across diverse DNN training workloads that common choices of batch size can lead to more energy consumption for the same target accuracy.}
Specifically, we performed a sweep over a large range of valid batch sizes (from 8 to the maximum batch size that fits in GPU memory)
for six \revised{deep learning workloads including} computer vision (CV), natural language processing (NLP), recommendation, and speech \revised{recognition} on an NVIDIA V100 GPU (Figure~\ref{fig:eta-potential}).\footnote{We measure GPU power consumption \revised{using} NVML~\cite{nvml}.}
Section~\ref{sec:methodology} provides details on workloads and methodology.
We find that the energy-optimal batch size (Batch Size Opt. in Figure~\ref{fig:eta-potential}) can lead to {3.4\%--65.0\%} lower energy consumption than the \revised{default choice} for the same target accuracy.
\begin{figure}[!t]
\centering
\includegraphics[width=0.85\linewidth]{Figures/motivation/eta-potential-all-v100.pdf}
\vspace*{-2mm}
\caption{Energy usage normalized against baseline for DNN training, measured on NVIDIA V100 GPU. Baseline uses maximum power limit and \revised{the default batch size presented in the original model publication when available or the maximum batch size which can consistently reach the target metric.}}
\label{fig:eta-potential}
\end{figure}
\begin{figure*}[!t]
\centering
\subfloat[][Energy-Time Tradeoff]{
\label{fig:eta-tta-with-bound}
\includegraphics[width=0.382\linewidth]{Figures/model/pareto-annotated-librispeech.pdf}
}
\hfil
\subfloat[][Pareto Front Zoom-in]{
\label{fig:eta-tta-zoom-in}
\includegraphics[width=0.382\linewidth]{Figures/model/pareto-librispeech.pdf}
}
\caption{DeepSpeech2 trained with LibriSpeech on NVIDIA V100: (a) ETA vs. TTA. The red dots indicate all feasible configurations. The two gray dotted lines indicate two boundaries characterized by average power consumption. The green line indicates the Pareto frontier over all configurations. (b) Zoom-in view on the Pareto frontier \revised{in (a)}, with batch size and power limit annotated on each data point.}
\label{fig:eta-tta-tradeoff}
\end{figure*}
\paragraph{Impact of GPU power limit on energy efficiency.}
\revised{Setting a GPU's power limit will have the device internally trigger dynamic voltage and frequency scaling (DVFS) such that its power draw does not exceed the power limit~\cite{dvfs-survey}.}
\revised{If not set manually, the power limit is at the maximum by default.}
We performed a sweep over a wide range of GPU power limits\footnote{From the minimum to the maximum power limit allowed by NVIDIA System Management Interface~\cite{nvidia-smi}; from 100W to 250W for NVIDIA V100.} for the aforementioned setup.
We found that the optimal energy consumption (Power Limit Opt. in Figure~\ref{fig:eta-potential}) may happen at a lower power limit than the maximum and can reduce energy consumption by {3.0\%--31.5\%}.
\paragraph{Joint optimization.}
As Figure~\ref{fig:eta-potential} shows, we can achieve even more energy savings ({23.8\%--74.7\%} reduction) if we jointly optimize both configurations.
Note that we observed similar opportunities for reducing energy consumption for other generations of GPUs as well (Figure~\ref{fig:appendix-eta-potential} in Appendix~\ref{sec:appendix-eta-potential}).
\subsection{Energy-Performance Tradeoffs}
\label{sec:motivation-pareto}
Opportunities for reducing DNN training energy consumption comes with a cost.
When optimized for \revised{energy} efficiency, DNN training performance (\revised{time-to-accuracy, or} TTA) may be impacted.
In the following, we characterize \revised{this} tradeoff.
We define the \revised{energy consumption of DNN training until it reaches its target accuracy} as its \emph{energy-to-accuracy} (ETA):
\begin{equation}
\begin{split}
\label{eq:eta-tta-power}
\mathtt{ETA}(b,p) & = \mathtt{TTA}(b,p) \times \mathtt{AvgPower}(b,p) ,
\end{split}
\end{equation}
where $p$ denotes the GPU power limit, $b$ the batch size, and $\mathtt{AvgPower}(b,p)$ the average power consumption during training with configuration $(b,p)$.
\revised{Similar to TTA, ETA captures the end-to-end goal of DNN training.}
\revised{Note that $\mathtt{AvgPower}(b,p)$ is not the same as the GPU power limit.}
\revised{When changes in configuration $(b,p)$ lead to an increase in TTA, ETA does not always follow because $\mathtt{AvgPower}(b,p)$ can decrease more.}
\revised{This motivates us to investigate the \emph{tradeoff}} between ETA and TTA.
\paragraph{Tradeoff between ETA and TTA.}
We characterize and elaborate on this tradeoff using DeepSpeech2 \revised{trained} on LibriSpeech as an example (Figure~\ref{fig:eta-tta-tradeoff}).
It shows a scatter plot of (TTA, ETA) for the \revised{batch size and power limit sweep} experiments in Section~\ref{sec:motivation-eta-gap}.
We observe similar results for other workloads as well (Figure~\ref{fig:appendix-eta-tta-pareto} in Appendix~\ref{sec:appendix-eta-tta}).
Let us start with Figure~\ref{fig:eta-tta-with-bound}, where each data point denotes the (TTA, ETA) of training the model for a certain configuration
While sweeping the configurations, we focus on the boundary of all feasible (TTA, ETA) pairs.
We find them to be bounded by two straight lines characterizing the average GPU power consumption.
\revised{When the GPU is under heavy load, the (TTA, ETA) data points appear closer to 210W.}
\revised{On the other hand, when the GPU is under lighter load, its average power consumption tends closer to 90W, which is close to the GPU's idle power consumption of 70W.}
More importantly, we find a curve along which all (TTA, ETA) pairs achieves Pareto optimality~\cite{censor1977pareto}, for which we cannot improve ETA without sacrificing TTA, and vice versa.
Now let us take a closer look at the Pareto frontier in Figure~\ref{fig:eta-tta-zoom-in}, with the configurations used during training annotated along each data point.
We highlight two takeaways:
\begin{denseenum}
\item These results \revised{show that} baseline configurations can lead to suboptimal energy efficiency (\S\ref{sec:motivation}).
Moreover, it shows that blindly going for high batch size and power limit configurations can lead to suboptimal TTA as well.
\item There exists a tradeoff between ETA and TTA, with different optimums for each.
The configuration optimizing the ETA ($b=$32, $p=$100W) is different from that optimizing TTA ($b=$48, $p=$250W).
\end{denseenum}
\section{Conclusion}
In this work, we sought to understand and optimize the energy consumption of DNN training on GPUs.
We identified the tradeoff between energy consumption and training time, and demonstrated that common practices can lead to inefficient energy usage.
{Zeus\xspace} is an online optimization framework for recurring DNN training jobs that \emph{finds} the Pareto frontier and allows users to \emph{navigate} the frontier by automatically tuning the batch size and GPU power limit of their jobs.
{Zeus\xspace} outperforms the state-of-the-art in terms of energy usage for diverse workloads and real cluster traces by continuously adapting to dynamic workload changes such as data drift.
We earnestly hope that {Zeus\xspace} will inspire the community to consider energy as a first-class resource in DNN optimization.
\section{{Zeus\xspace} Overview}
\label{sec:overview}
{Zeus\xspace} is an optimization framework that navigates the ETA-TTA tradeoff by automatically configuring the batch size and GPU power limit of recurring DNN training jobs.
It enables developers to optimize energy and/or performance metrics using a single knob.
\subsection{Optimization Metric}
\label{sec:hybrid-cost-metric}
Defining a good cost metric for \revised{users} to express their \revised{preference} in this tradeoff is critical in designing {Zeus\xspace}.
We propose a simple cost metric:
\begin{equation}
\label{eq:cost-metric}
C(b,p;\revised{\eta}) = \eta \cdot \mathtt{ETA}(b,p) + (1-\eta) \cdot \mathtt{MAXPOWER} \cdot \mathtt{TTA} (b,p)
\end{equation}
Here $\eta$ is \revised{the parameter specified by the user to express} the relative importance of energy efficiency and training performance (throughput).
When $\eta=0$, we are only optimizing for time consumption, whereas when $\eta=1$, we are only optimizing for energy consumption.
$\mathtt{MAXPOWER}$ is the maximum power limit supported by the GPU, a constant introduced to unify the units of measure in the cost metric.
\subsection{Challenges in Picking the Optimal Configuration}
Combining Equations~\ref{eq:eta-tta-power} and~\ref{eq:cost-metric}, we have:
\begin{equation}
\label{eq:cost-tta-power}
C = \left(\eta \cdot \mathtt{AvgPower}(b,p) + (1-\eta) \cdot \mathtt{MAXPOWER}\right) \cdot \mathtt{TTA} (b,p).
\end{equation}
Picking the optimal configuration(s) to minimize the energy-time cost $C$ for DNN training is challenging because the search space $[b \times p]$ is large and obtaining the cost of each configuration is difficult.
\revised{This is because it is hard to determine the value of both $\mathtt{AvgPower}(b,p)$ and $\mathtt{TTA}(b,p)$ efficiently, as explained below.}
\begin{denseitemize}
\item \textbf{Complex power consumption model:} The total energy consumption of a GPU is affected in a non-linear fashion by both the characteristics of the workload such as the number of instructions and memory accesses, as well as the GPU hardware configurations such as the frequency and voltage of the cores and memory on board~\cite{accelwattch,verified-gpu-instruction-energy-model}.
\revised{Existing efforts estimate GPU energy consumption based on instruction- or kernel-level information~\cite{integrated-gpu-power-model, gpu-power-model-analytical}, which are architecture-specific and workload-dependent.}
\item \textbf{Stochastic nature of DNN training:} Modeling and predicting the duration for training a specific model to target accuracy (TTA) is known to be difficult~\cite{tiresias}.
Moreover, \revised{the} randomness introduced during model initialization and data loading leads to variations of TTA, even when the same job is run on the same GPU with the same configuration -- TTA variations can be as large as 14\%~\cite{dawnbench-tta}.
\end{denseitemize}
Fortunately, DNN training jobs often recur in production \revised{clusters}~\cite{applied-ml-at-fb,alibaba-trace}.
This provides opportunities for empirical estimation through repeated measurements across recurrences of the same training job.
\subsection{Architectural Overview}
At a high-level, {Zeus\xspace} takes an online exploration-exploitation approach to minimize the aggregate cost of recurrent DNN training jobs.
{Zeus\xspace} addresses the aforementioned challenges with two key components:
\begin{denseenum}
\item A just-in-time (JIT) online profiler, which efficiently profiles the energy characteristics of the training job online.
\item Multi-Armed Bandit (MAB) with Thompson sampling, which allows us to embrace the stochastic nature of DL training and optimize under uncertainty while also adapting to changing workloads such as data drift.
\end{denseenum}
\revised{The combination of the JIT profiler and MAB makes {Zeus\xspace} a fully online solution, allowing it to immediately begin optimizing for incoming jobs.}
\begin{figure}
\centering
\includegraphics[trim=300 110 320 75,clip,width=0.795\linewidth]{Figures/design/architecture.pdf}
\caption{{Zeus\xspace} Workflow.}
\label{fig:system-overview}
\end{figure}
\paragraph{Workflow of {Zeus\xspace}.}
Figure~\ref{fig:system-overview} shows an overview of the high-level workflow of {Zeus\xspace}.
In a production environment, users submit \blackcircled{1} recurrent DNN training jobs (a tuple of data, model, optimizer, and the target \revised{validation metric}) to {Zeus\xspace}, along with a set of feasible batch sizes $\mathcal{B}$ and power limits $\mathcal{P}$ to explore.
{Zeus\xspace} then predicts \blackcircled{2} the optimal batch size and power limit configuration based on past execution history, and launches \blackcircled{3} the training job with \revised{that} configuration.
During and after the training process, \blackcircled{4} statistics about DNN training (e.g., validation \revised{metric}) and GPU power consumption are collected and fed back to the {Zeus\xspace} optimizer.
The {Zeus\xspace} optimizer learns from the feedback and adjusts its internal states.
The training job will be terminated upon either reaching target \revised{metric} or exceeding a stopping threshold \revised{determined by {Zeus\xspace}}.
The whole process is an automated feedback loop that minimizes the key objective of energy-time cost.
Building {Zeus\xspace} requires both algorithm design and systems support.
Next we describe the core optimization algorithm details (\S\ref{sec:design}) and {Zeus\xspace} implementation highlights (\S\ref{sec:impl}).
\section{Related Work}
\paragraph{DNN training.}
A large body of recent studies focus on creating fast kernels for tensor operations~\cite{taso,PET,ansor,tvm}, efficiently placing data and/or computation~\cite{pipedream,zero,gspmd,oort}, and optimizing communication~\cite{blinkml,bytescheduler}.
However, most of them optimize for TTA and are oblivious of their energy impact.
These works can be applied together with {Zeus\xspace}, potentially accelerating training while making it energy efficient.
Another recent effort in reducing TTA (without considering energy) in multi-GPU DNN training \revised{settings} is Pollux~\cite{pollux}.
Pollux dynamically changes the batch size \emph{during} training based on the Gradient Noise Scale (GNS)~\cite{gns}.
However, GNS does not theoretically capture the generalization of the model~\cite{gns} and can only be efficiently approximated when there are more than one GPUs participating in training.
{Zeus\xspace}, on the other hand, optimizes and trades off TTA and ETA by tuning the batch size \emph{across} job recurrences and does not alter the model's convergence characteristics.
\paragraph{Energy measurement for Deep Learning.}
A recent line of research has analyzed the energy consumption~\cite{patterson2021carbon} as well as the environmental impact~\cite{energy-nlp-policy, quantify-carbon} for training large DNN models inside a cluster.
On the device-level, benchmarking efforts have been made to understand the energy efficiency and performance of training DNN on GPUs and other accelerators~\cite{benchmark-ai-accelerators}.
Several Python frameworks have been built for measurement~\cite{experiment-impact-tracker, build-ontopof-impact-tracker} and prediction~\cite{carbontracker} of energy consumption for DNN training.
{Zeus\xspace} takes a similar software-based approach to measure power consumption via NVML~\cite{nvml}, in order to perform JIT profiling of DNN training jobs.
\paragraph{Energy optimization for Deep Learning.}
Existing work has investigated energy-accuracy tradeoff in the context of DNN inference with new neural network architecture~\cite{alert} and algorithm-hardware co-design~\cite{edgebert}, and training strategies such as warm-start~\cite{warmstart-training} and gradient-matching-based data subset selection~\cite{gradmatch-icml}.
Other works optimize energy for DNN training on multiple GPUs with scheduling~\cite{energy-clusterman} and task mapping~\cite{dvfs-taskmap-powercap}.
{Zeus\xspace} complements these solutions as it can be plugged in transparently into these frameworks.
Several works have studied the impact of GPU \revised{dynamic frequency and voltage scaling (DVFS)} and power configuration on the energy consumption and performance of DNN training~\cite{odpp, gpoeo, dvfs-impact, dynamic-underclock-gpu,dvfs-taskmap-powercap}, wherein they focus on the tradeoff between the transient metric of system throughput and power consumption.
\revised{While these work rely on offline modeling and profiling, {Zeus\xspace} focuses on a more realistic end-to-end metric of energy-to-accuracy and is fully online.}
BatchSizer~\cite{batchsizer} introduces batch size as a control knob to optimize for energy efficiency of DNN inference.
{Zeus\xspace} focuses on DNN training, and takes a holistic approach, optimizing both GPU and job configurations together.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 563 |
\section{\label{sec:1}Introduction}
The polarization degree of freedom of single photons has been widely used to explore quantum phenomena, since polarization entangled photon pairs can be prepared with high fidelity and the polarization state is easy to manipulate with linear optics. For example, polarization entangled photons have been utilized for a variety of fundamental tests of quantum physics, including local hidden variables theories \cite{Kwiat95} and epistemic models of the wavefunction \cite{Ringbauer15}. In addition, a plethora of quantum information technologies have been experimentally implemented using photon polarization such as quantum key distribution \cite{Schmitt07}, quantum dense coding \cite{Mattle96}, quantum teleportation \cite{Kim01}, and quantum computing \cite{Lopez12}.
The polarization state of light is conventionally described through the use of Stokes parameters that can be represented as a polarization direction and a degree of polarization on the Poincar{\'e} sphere \cite{Stokes52}. Since the Stokes parameters show only the averaged, ``classical'' features, they are not sufficient to describe the quantum polarization features fully. For instance, there exist ``classically'' unpolarized light which has non-isotropic second-order polarization, thus making the state polarized \cite{Usachev01}. This example highlights the existence of hidden polarization features and the importance of polarization fluctuations. Up to now, various quantum states have been studied, such as, squeezed polarization states \cite{Shalm09,Klimov10} and entangled photon states \cite{Jaeger03}. In addition, efficient polarization tomography methods have been suggested \cite{Schilling10,Soderholm12,Bjork12,Bayraktar16}.
In this paper, we experimentally investigate various quantum polarization features of three-photon quantum states, including product and entangled states with varying purity. The studied three-photon states are isomorphic to the states of a composite system consisting of three spin-1/2 particles with the bosonic characteristic, that is, the symmetric Hilbert subspace. To fully describe the properties of the states, up to the third-order Stokes parameters are necessary and sufficient, since the fourth- and higher-order polarization moments contain no additional information. The central moments of the Stokes operator is also useful to describe the polarization distribution on the Poincar{\'e} sphere with Gaussian approximation \cite{Bjork12}. The mean, variance, skewness represent the first, second, third order central moments, respectively. Through their central moments, all three-photon quantum states are categorized into six different classes according to Table~\ref{tab:table1} based on their rotation invariance (on the Poincar{\'e} sphere) of their the mean, variance and skewness. In this work, six class-representative three-photon quantum states are experimentally prepared and measured to confirm the predicted polarization properties.
\renewcommand{\arraystretch}{1.5}
\begin{table*}
\caption{\label{tab:table1}
Classification and examples of three-photon quantum polarization based on SU(2) rotation invariance of mean $\langle \hat{S}_{\mathbf{n}} \rangle$, variance $\langle \hat{\Delta}^{2}_{\mathbf{n}} \rangle$, and skewness $\langle \hat{\Delta}^{3}_{\mathbf{n}} \rangle$ \cite{Bjork12}. The symbol O indicates rotation invariance of the particular order of quantum polarization.}
\begin{tabular}{>{\centering}m{.1\linewidth}>{\centering}m{.1\linewidth}>{\centering}m{.1\linewidth}m{.05\linewidth}m{.4\linewidth}}
\hline\hline
\multicolumn{3}{c}{Rotation invariance\footnote{For three-photon polarization states, six classes are physically possible among eight possibilities.}}&\\ \cline{1-3}
$\langle \hat{S}_{\mathbf{n}} \rangle$&$\langle \hat{\Delta}^{2}_{\mathbf{n}} \rangle$&$\langle \hat{\Delta}^{3}_{\mathbf{n}} \rangle$&&\multicolumn{1}{c}{\ \ Representative state in each class\footnote{Fock states in horizontal and vertical polarization mode.}}\\ \hline
O&O&O&&$\ \ \ \ \hat{\mathbb{I}}/4$\\
O&O&X&&$\ \ \ \ \frac{1}{3}\!\left |3,0\rangle\langle 3,0 \right |+\frac{1}{2}\!\left | 1,2\rangle\langle 1,2 \right |+\frac{1}{6}\!\left | 0,3\rangle\langle 0,3 \right |$\\
O&X&O&&$\ \ \ \ \frac{1}{2}\!\left ( \left | 3,0\rangle\langle 3,0 \right |+\left | 0,3\rangle\langle 0,3 \right |\right )$\\
O&X&X&&$\ \ \ \ \frac{1}{\sqrt{2}}\!\left ( | 3,0\rangle-i| 0,3\rangle\right )$\\
X&O&X&&$\ \ \ \ \frac{19}{36}\!\left |3,0\rangle\langle 3,0 \right |+\frac{15}{36}\!\left | 1,2\rangle\langle 1,2 \right |+\frac{1}{18}\!\left | 0,3\rangle\langle 0,3 \right |$\\
X&X&X&&$\ \ \ \ | 3,0\rangle$\\
\hline\hline
\end{tabular}
\end{table*}
\section{\label{sec:2}Theory}
The Stokes parameters consist of the total intensity $S_{0}$ and the three elements of Stokes vector $\vec{S}=(S_{1},S_{2},S_{3})$ which represent complementary polarization directions on the Poincar{\'e} (or Bloch) sphere. The values of $(S_{1},S_{2},S_{3})$ are obtained from intensity differences between orthogonal polarizations: diagonal/anti-diagonal, right/left circular, and horizontal/vertical polarizations, respectively. By substituting the bosonic number operator for intensity, the Stokes operators can be well defined and they give quantized values of the Stokes parameters. The operators are expressed as \cite{Usachev01}
\begin{equation}
\begin{split}\label{eq:01}
&\hat{S}_{0}=\hat{a}^{\dagger}_{H}\hat{a}_{H}+\hat{a}^{\dagger}_{V}\hat{a}_{V}, \ \ \ \ \ \ \hat{S}_{1}=\hat{a}_{H}\hat{a}^{\dagger}_{V}+\hat{a}^{\dagger}_{H}\hat{a}_{V}, \\
&\hat{S}_{2}=i(\hat{a}_{H}\hat{a}^{\dagger}_{V}-\hat{a}^{\dagger}_{H}\hat{a}_{V}),\ \ \ \hat{S}_{3}=\hat{a}^{\dagger}_{H}\hat{a}_{H}-\hat{a}^{\dagger}_{V}\hat{a}_{V}
\end{split}
\end{equation}
where $\hat{a}_{H}$ $(\hat{a}_{V})$ is the annihilation operator for the horizontal (vertical) polarization mode. Their commutation relationships can be derived from the bosonic commutation relations between annihilation operators.
\begin{subequations}
\begin{eqnarray}
&[&\hat{S}_{0},\hat{S}_{j}]=0,\label{eq:02a}\\
&[&\hat{S}_{j},\hat{S}_{k}]=i2\epsilon_{jkl}\hat{S}_{l}, \ \ \ \ \ \ j,k,l \in \{1,2,3\} \label{eq:02b}
\end{eqnarray}
\end{subequations}
where $\epsilon_{jkl}$ is the Levi-Civita tensor symbol.
The physical properties of the Stokes operators are implied by the commutation relations. As indicated by Eq.~(\ref{eq:02a}), the commutation between the total photon number operator $\hat{S}_{0}$ and all other Stokes operators $\hat{S}_{j}$, indicates that the Stokes parameters and photon number can be measured independently without mutual disturbance. This allows measurements of the Stokes parameters for specific photon-number states with photon-number resolving detectors. This also implies that any moment of the Stokes operators $\hat{S}_{1}$, $\hat{S}_{2}$, and $\hat{S}_{3}$ can be measured in a similar manner.
Moreover, Eq.~(\ref{eq:02b}) indicates that some Stokes parameters must be uncertain, leading to the following inequalities,
\begin{subequations}
\begin{eqnarray}
&\sqrt{\langle\hat{\Delta}_{j}^2 \rangle}\sqrt{\langle \hat{\Delta}_{k}^2\rangle}\geq\left | \epsilon_{jkl}\langle \hat{S}_{l} \rangle \right |,\ \ \ j,k,l \in \{1,2,3\}\label{eq:03a}\\
&2\langle\hat{S}_{0}\rangle\leq \langle\hat{\Delta}_{1}^2\rangle+\langle\hat{\Delta}_{2}^2\rangle+\langle\hat{\Delta}_{3}^2\rangle \leq \langle \hat{S}_{0} \rangle (\langle \hat{S}_{0}\rangle+2)\label{eq:03b}
\end{eqnarray}
\end{subequations}
where $\Delta_{j} \equiv \hat{S}_{j}-\langle \hat{S}_{j} \rangle$ is the central-moment of the Stokes operator $j$. The existence of a photonic quantum state with hidden polarization, i.e., high-order central-moments may have non-zero values even though the state may be first-order unpolarized, $\langle \hat{\vec{S}} \rangle=0$, is implied in Eq.~(\ref{eq:03a}). Note that, Eq.~(\ref{eq:03b}) gives boundaries of the sum of the second-order central moments. For three-photon quantum states, the sum is bounded between 6 and 15.
The notion of Stokes operators can be generalized by defining
\begin{equation}
\hat{S}_{\mathbf{n}} = (\hat{S}_{1},\hat{S}_{2},\hat{S}_{3}) \cdot \mathbf{n},
\end{equation}
where $\mathbf{n}$ is a unit vector on the Poincar\'{e} sphere. The operator $\hat{S}_{\mathbf{n}}$ assesses the polarization state of the photonic quantum state in the direction $\mathbf{n}$. It follows trivially that one can define polarization of order $m$ in direction $\mathbf{n}$ as $\langle \Delta_{\mathbf{n}}^m \rangle$ where $\Delta_{\mathbf{n}} \equiv \hat{S}_{\mathbf{n}}-\langle \hat{S}_{\mathbf{n}} \rangle$.
\section{\label{sec:3}Experiment}
To confirm the quantum polarization features of three-photon quantum states experimentally, six representative three-photon quantum states are prepared, as shown in Table~\ref{tab:table1}. The three-photon states are generated from the double pair emission of femtosecond-pulse-pumped spontaneous parametric down-conversion (SPDC), see Fig.~\ref{fig:fig1} \cite{Kim09,Kim11}. The pump pulse is derived from a frequency-doubled mode-locked Ti:Sapphire laser and has a central wavelength of 390 nm, a pulse duration of 140 fs, and a repetition rate of 80 MHz. The SPDC photons, generated from a type-I BBO crystal, have a central wavelength of 780 nm and propagates non-collinearly with the pump laser.
To prepare a heralded three-photon quantum state via interference using the scheme in Fig.~\ref{fig:fig1}, it is essential that the inherent frequency correlation between the SPDC photon pair be eliminated \cite{Grice01}. In our work, we ensure that this condition is satisfied by using a 0.6 mm thick type-I BBO crystal, generating broadband SPDC photons, and by filtering the SPDC photons with interference filters (IF) with full width at half maximum (FWHM) of 3 nm. The calculated spectral properties of the unfiltered SPDC photons are shown in Fig.~\ref{fig:fig2}(a). It is clear that the SPDC photons from a 0.6 mm thick type-I BBO has a very broadband emission and show a very strong spectral correlation between the pair. When the SPDC photons are filtered with the 3 nm interference filters, see Fig.~\ref{fig:fig2}(b), the resulting SPDC photons have almost no spectral correlations, see Fig.~\ref{fig:fig2}(c).
\begin{figure}[tp]
\centering
\includegraphics[width=3.4in]{fig1_exp_setup.pdf}
\caption{Experimental scheme for generation and measurement of three-photon states. Two pairs of SPDC photons are sent to an interferometer through single mode fibers (SMF) after passing 3 nm bandwidth interference filters (IF). In the state preparation interferometer, the target states are prepared with a polarizing beam splitter (PBS), a partially-polarizing beam splitter (PPBS), half- and quarter-wave plates (HWP, QWP), and a linear polarizer (LP). Conditioned on the detection of a single photon at detector D1, three photons are prepared in mode $b$ in a particular quantum state set by QWP1, HWP2, and LP. Four-fold coincidence measurements with detectors D1, D2, D3, and D4 for sixteen polarization projection measurement allow quantum state tomography for the heralded three-photon states.
}\label{fig:fig1}
\end{figure}
The non-collinear, double pair SPDC photons are combined into a single spatial mode by a PBS through single-mode optical fibers for spatial mode cleaning, see Fig.~\ref{fig:fig1}. Initially, all four photons are horizontally polarized but the use of a fiber polarization controller allows us to combine all four photons to a single spatial mode without loss. As all four photons must also be indistinguishable temporally, they all need to arrive at PBS simultaneously. This has been achieved by observing the Shih-Alley/Hong-Ou-Mandel dip between the SPDC photon at the PBS \cite{Shih86,Hong87}. To observe the two-photon interference dip, the angles of HWP1 and HWP3 are set, respectively, at 22.5$\degree$ and 45$\degree$. All other wave plates, HWP2, QWP1 and QWP2, are set at 0$\degree$ and LP is removed. the coincidence between the detectors D1 and D2 are measured by moving the translation stage on horizontal input mode in Fig.~\ref{fig:fig1}. The experimental result shown in Fig.~\ref{fig:fig2}(d) exhibits the dip visibility of 95.0\% at 260 mW pump power (99.6\% after multi-photon noise subtraction). The translation stage is then set so that the SPDC photons are arriving at the PBS simultaneously. Then, the four-photon quantum state, resulting from the double-pair emission of the SPDC, after the PBS is written as $|2,2\rangle_{H,V}$.
\begin{figure}[tp]
\centering
\includegraphics[width=3.4in]{fig2_JSI_HOM.pdf}
\caption{The simulated joint spectra of (a) SPDC photons, (b) interference filters with 3 nm FWHM bandwidth, and (c) SPDC photons filtered with the interference filters. The simulated joint spectrum in (c) shows that the spectral correlation between the photon pair is well eliminated by the interference filters. (d) The Shih-Alley/Hong-Ou-Mandel interference dip has the visibility of 95.0\% (99.6\% after multi-photon noise subtraction) at 260 mW pump power, a clear experimental indication that spectral distinguishability between the SPDC photons have been well eliminated by the interference filters. The red solid line is the Gaussian fit to the data. The blue solid line is the linear fit to the calculated multi-photon noise.
}\label{fig:fig2}
\end{figure}
We now describe the scheme for heralding a three-photon quantum state in mode $b$ by detecting a single photon at D1. The initial four-photon state $|2,2\rangle_{H,V}$ passes through HWP1 and the HWP1 angle (0$\degree$ or 22.5$\degree$) is set differently for preparing different three-photon states. Specifically, HWP1 is set at 0$\degree$ to prepare $|1,2\rangle$ and $|2,1\rangle$ and set at 22.5$\degree$ to prepare $\frac{1}{\sqrt{2}}\!(|3,0\rangle-i|0,3\rangle), |3,0\rangle$, and $|0,3\rangle$. After HWP1, the state becomes
\begin{subequations}
\begin{eqnarray}
&|&H\!W\!\!P1\rangle_{0\degree} \ \ = \frac{1}{2}{a^{\dagger\ 2}_H}{a^{\dagger\ 2}_V}|0\rangle,\label{eq:04a}\\
&|&H\!W\!\!P1\rangle_{22.5\degree} \!=\!\!\left (\frac{1}{8}{a^{\dagger\ 4}_H}-\frac{1}{4}{a^{\dagger\ 2}_H}{a^{\dagger\ 2}_V}+\frac{1}{8}{a^{\dagger\ 4}_V}\right )\!|0\rangle.\label{eq:04b}
\end{eqnarray}
\end{subequations}
The subscripts 0$\degree$ and 22.5$\degree$ indicate the angles of HWP1. The photons then impinge on the partially-polarizing beam splitter (PPBS) designed for unity reflection for vertical polarization and 1/3 partial reflection for horizontal polarization. Considering the case when one photon is transmitted and found in mode $a$ and three photons are reflected by the PPBS and found in mode $b$, the reflected three-photon state heralded by the presence of a single-photon in mode $a$ is given by
\begin{subequations}
\begin{eqnarray}
&|&P\!P\!B\!S\rangle_{0\degree}^b \ \ = \frac{1}{\sqrt{2}}{a^{\dagger}_H}{a^{\dagger\ 2}_V}|0\rangle,\label{eq:05a}\\
&|&P\!P\!B\!S\rangle_{22.5\degree}^b \!=\!\!\left ( \frac{1}{9\sqrt2}e^{2i\phi}{a^{\dagger\ 3}_H}-\frac{1}{3\sqrt{2}}{a^{\dagger}_H}{a^{\dagger\ 2}_V}\right )\!|0\rangle,
\label{eq:05b}
\end{eqnarray}
\end{subequations}
where the phase $\phi$ comes from the relative phase-difference between the two orthogonal polarizations when they are reflected at the PPBS.
\begin{figure*}[tp]
\centering
\includegraphics[width=6.5in]{fig3_density_matrix.pdf}
\caption{Density matrices of experimentally prepared heralded three-photon states. Bold lines in the density matrices indicate ideal target density matrices. The fidelity is calculated between the ideal target matrix and the experimentally generated density matrix.
}\label{fig:fig3}
\end{figure*}
The transmitted photon in mode $a$ is used for heralding of the other three photons by a ``click'' at detector D1. Then, after the photons pass through QWP1 and HWP2, either both set at 0$\degree$ for Eq.~(\ref{eq:06a})
or QWP1 at $45\degree$ and HWP2 at $\phi/4$ for Eq.~(\ref{eq:06b}), the heralded three-photon state becomes,
\begin{subequations}
\begin{eqnarray}
&|&H\!W\!\!P2\rangle_{0\degree} \ \ =\frac{1}{\sqrt{2}}{a^{\dagger}_H}{a^{\dagger\ 2}_V}|0\rangle\label{eq:06a},\\
&|&H\!W\!\!P2\rangle_{22.5\degree} \!=\frac{1}{2\sqrt{3}}\left ({a^{\dagger\ 3}_H}-i{a^{\dagger\ 3}_V}\right )\!|0\rangle.\label{eq:06b}
\end{eqnarray}
\end{subequations}
Here, we see that the heralded three photon states $|1,2\rangle$ and $\frac{1}{\sqrt{2}}\!(|3,0\rangle-i|0,3\rangle)$ have been prepared. The state $|2,1\rangle$ can be prepared from $|1,2\rangle$ with the help of HWP2 set at 45$\degree$. Also, the states $|3,0\rangle$ and $|0,3\rangle$ can be post-selected from the entangled state or the N00N state $\frac{1}{\sqrt{2}}\!(|3,0\rangle-i|0,3\rangle)$ with a linear polarizer (LP).
We first measure the the value of $\phi$ experimentally by using the $|1,1\rangle_{H,V}$ component of the SPDC leading to the coincidence event between detectors D1 and D2. First, HWP1 and QWP1 are set at 15$\degree$ and $45\degree$, respectively. The photon in mode $b$ is measured on the projection basis $\frac{1}{\sqrt{2}}(|H\rangle - |V\rangle)$. Then, as a function of the HWP2 angle $\theta$, the coincidence count between detectors D1 and D2 will be proportional to $\sin^{2}\{\frac{1}{2}(\phi-4\theta)\}$. Thus, by measuring the angle $\theta$ at which the coincidence count is minimized, it is possible to determine the phase $\phi$. Experimentally, we find that $\phi=-85.7\degree$.
In our scheme, the double-pair event of SPDC contributes to the heralded three-photon state but triple-pair or higher order SPDC events lead to multi-photon noise as they can also trigger the four-photon coincidence circuit. As $N$-pair events of SPDC with $N \geq 3$ increase with the double-pair event of SPDC, one needs to consider the trade-off between the detection rate and the multi-photon noise contribution to the data. In our setup, we find that 260 mW pump power results in 2.5\%, 0.06\%, and 0.002\% emission probabilities of a single, double, triplet pairs, respectively. In case of the states $|3,0\rangle$ and $|0,3\rangle$, on the other hand, the four-fold coincidence probability is half of the state $\frac{1}{\sqrt{2}}\!(|3,0\rangle-i|0,3\rangle)$. Thus, we are able to use twice high pump power in this case to reduce the measurement time.
Finally, the heralded three-photon state is characterized by performing quantum state tomography using 16 projection measurements set by QWP2, HWP3, and PBS in Fig. \ref{fig:fig1} and maximum likelihood estimation \cite{Israel12}. The experimentally obtained density matrices for the representative three-photon states listed in Table I are shown in Fig.~\ref{fig:fig3}. Mixed states are obtained by incoherently adding pure states with the proper ratio. The three-photon entangled state $\frac{1}{\sqrt{2}}\!(|3,0\rangle-i|0,3\rangle)$ has the lowest fidelity because the triple-pair contribution from pulsed SPDC to the non-interfering background is more noticeable \cite{Kim09,Kim11}.
\section{\label{sec:4}Analysis}
In this section, we analyze the experimentally generated three-photon polarization states with the central-moment description of quantum polarization up to the third order. (The fourth and higher order Stokes moments can be expressed as function of the lower order moments.) The central-moments, plotted in the corresponding direction on the Poincar{\'e} sphere, shows quantum polarization features such as rotation invariants and the polarization uncertainty distribution. For instance, classical unpolarized light is SU(2) rotation invariant, that is, it is rotationally invariant in all directions on the sphere. If the first $\mathit{m}$ moments are rotationally invariant, the state is defined to be $\mathit{m}$-th order unpolarized \cite{Bjork12}. As for the uncertainty relation regarding quantum polarization in Eq.~(\ref{eq:03b}), all three-photon states have a common restriction that the sum of the variances along three orthogonal polarizations on the Poincar{\'e} sphere (indicating three complementary polarization bases) is between 6, from $2\langle \hat{S_0}\rangle$, and 15, from $\langle \hat{S_0}\rangle (\langle \hat{S_0}\rangle+2))$ because $\langle \hat{S}_{0}\rangle= 3$.
The identity density matrix is invariant of any rotation transformation since $\hat{\mathbb{I}}/4=\hat{U}^{\dagger}_{\mathbf{n}}(\hat{\mathbb{I}}/4)\hat{U}_{\mathbf{n}}$ where $\hat{U}_{\mathbf{n}}$ is any SU(2) rotation. So, this state always gives an isotropic expectation value for quantum polarization in any basis. That is, every order of its quantum polarization is rotationally invariant and the state is hence unpolarized to every order. Since the state is first-order unpolarized, the second-order central-moment or the variance, is given by the mean square of the Stokes operator. As the eigenvalues of $\hat{S}_{3}$ for the states $|3,0\rangle, |2,1\rangle, |1,2\rangle$, and $|3,0\rangle$ are $3, 1, -1,$ and $-3$, respectively, the value $\langle \hat{\Delta}^{2}_{3}\rangle$ for the state $\hat{\mathbb{I}}/4$ is $(9+1+1+9)/4 =5$. This state has the isotropic variance, so the sum of variance is 15, making it the maximum sum-uncertainty state. As expected, in Fig.~\ref{fig:fig4} (a), the experimental result confirms the theoretical results with only small errors. Note that for the odd polarization orders, the illustrated quantity is the absolute value of the moment plotted as a function of the direction $\mathbf{n}$ on the Poincar\'{e} sphere. For odd polarization orders $m$ it holds that $\langle \Delta_{\mathbf{n}}^m \rangle = -\langle \Delta_{\mathbf{-n}}^m \rangle$.
Any second-order unpolarized three-photon quantum state must be a mixed state, see the proof in \cite{Bjork12}. Among the mixed states, the state $\frac{1}{3}\!\left |3,0\rangle\langle 3,0 \right |+\frac{1}{2}\!\left | 1,2\rangle\langle 1,2 \right |+\frac{1}{6}\!\left | 0,3\rangle\langle 0,3 \right |$ has the maximum sum-uncertainty. This state is isotropic up to the second-order and is polarized in the third-order, as shown theoretically and experimentally in Fig.~\ref{fig:fig4}(b).
\begin{figure*}[tp]
\centering
\includegraphics[width=6.8in]{fig4_quantum_pol.pdf}
\caption{Central-moment quantum polarization descriptions of assorted three-photon states. The rainbow-colored gradient represents the distance from the origin of the coordinate system. In general, the red color illustrates the farther distance from the origin. Depicted in each panel are, left-to-right, the mean value $\langle |\hat{S}_{\mathbf{n}}|\rangle$, the variance $\langle \hat{\Delta}^{2}_{\mathbf{n}} \rangle$, and the absolute value of the skewness $\langle |\hat{\Delta}^{3}_{\mathbf{n}}| \rangle$). Note that, for odd polarization orders $m$, $\langle \Delta_{\mathbf{n}}^m \rangle = -\langle \Delta_{\mathbf{-n}}^m \rangle$ and, thus, the illustrated quantity is the absolute value of the moment plotted as a function of the direction $\mathbf{n}$ on the Poincar\'{e} sphere. }
\label{fig:fig4}
\end{figure*}
The states $|3,0\rangle$ and $|0,3\rangle$ have rotation symmetry about the $\hat{S}_{3}$ axis. The symmetry is preserved upon mixing (i.e., incoherent addition of quantum states) and they have vectors in opposite directions from each other on the Poincar{\'e} sphere. Therefore, the mixed state $\frac{1}{2}\! \left ( \left | 3,0\rangle\langle 3,0 \right |+\left | 0,3\rangle\langle 0,3 \right |\right )$ has vanishing all odd-order central moments as shown in Fig.~\ref{fig:fig4}(c). The second-order central moment is however maximized along the $\hat{S}_{3}$ axis (with the value of 9) and minimized on the $\hat{S}_{1}$-$\hat{S}_{2}$ plane (with the value of 3). The state $\frac{1}{2}\! \left ( \left | 3,0\rangle\langle 3,0 \right |+\left | 0,3\rangle\langle 0,3 \right |\right )$ therefore is a three-photon maximum sum-uncertainty state, having the quantum polarization properties of first-order hidden polarization, polarized in the second-order, and with no third-order polarization.
Let us now consider the case of the entangled state $\frac{1}{\sqrt{2}}\!\left ( | 3,0\rangle-i| 0,3\rangle\right )$. It is clear that some quantum polarization features of the entangled state would be similar to the mixed state $\frac{1}{2}\! \left ( \left | 3,0\rangle\langle 3,0 \right |+\left | 0,3\rangle\langle 0,3 \right |\right )$ as both contain the same basis states. This is reflected in the first-order and second-order quantum polarization properties. The key difference between the two states are inherent coherence and is reflected on the third-order quantum polarization, producing skewness in three directions as shown in Fig.~\ref{fig:fig4}(d). The three-photon N00N state is well-known to exhibit $N$ times phase sensitivity compared to a classical state and this feature is illustrated in the third-order quantum polarization, showing three oscillations during the 2$\pi$ phase change on the $\hat{S}_{1}$-$\hat{S}_{2}$ plane. Note that the figure shows the absolute value of central moment. In the experiment, the state is not quite ideal, see the density matrix in Fig.~\ref{fig:fig3}(d), so the skewness is somewhat reduced (resulting in reduced N00N state interference visibility) compared to the theoretical one. Note that the variances $\langle \Delta^{2}_{1} \rangle, \langle \Delta^{2}_{2} \rangle$, and $ \langle \Delta^{2}_{3} \rangle$ are calculated to be $3, 3,$ and $9$, respectively. Thus, the state is also a maximum sum-uncertainty state. The state has first-order hidden polarization but is polarized to second- and third-order.
We now consider the state $\frac{19}{36}\!\left |3,0\rangle\langle 3,0 \right |+\frac{15}{36}\!\left | 1,2\rangle\langle 1,2 \right |+\frac{1}{18}\!\left | 0,3\rangle\langle 0,3 \right |$ which is first-order and third-order polarized, but with isotropic second-order central moment, see Table I. The theoretical and experimental results shown in Fig.~\ref{fig:fig4}(e) illustrate this feature of the state. Note that, since the state is a mixture of horizontal and vertical basis eigenstates, it has rotational symmetry about the $\hat{S}_{3}$ axis. But, it is not a maximal sum-uncertainty state.
Finally, consider the state $| 3,0\rangle$ which is clearly first-order, second-order, and third-order polarized as all three photons are horizontally polarized and this is shown by the anisotropic features in all orders of central moments in Fig.~\ref{fig:fig4}(f). Since the state is an eigenstate of $\hat{S}_{3}$, $\langle\hat{\Delta}_{3}^{m}\rangle$ vanishes for all $m$. Moreover, both $\langle\hat{\Delta}_{1}^{m}\rangle$ and $\langle\hat{\Delta}_{2}^{m}\rangle$ vanish for odd $m$. Thus the odd polarization moments all vanish on the $\hat{S}_1$-$\hat{S}_2$ plane. Note that the state satisfies (\ref{eq:03a}) with equality on the $\hat{S}_{1}$-$\hat{S}_{2}$ plane.
The advantages of using the quantum polarization description of the multi-photon state can be summarized as follows. As evidenced in Fig.~\ref{fig:fig4}, the quantum polarization description allows one to visually identify for which applications the quantum state is best suited. For instance, the mixed state $\frac{1}{3}\!\left |3,0\rangle\langle 3,0 \right |+\frac{1}{2}\!\left | 1,2\rangle\langle 1,2 \right |+\frac{1}{6}\!\left | 0,3\rangle\langle 0,3 \right |$ shown in Fig.~\ref{fig:fig4}(b) can be useful for polarization interferometry involving three-photon correlation measurement. However, the mixed state $\frac{1}{2}\! \left ( \left | 3,0\rangle\langle 3,0 \right |+\left | 0,3\rangle\langle 0,3 \right |\right )$ shown in Fig.~\ref{fig:fig4}(c) is better suited for two-photon correlation interferometry due to the anisotropy in the second-order quantum polarization. Also, the N00N state offers the best phase sensitivity to SU(2) rotations, as evidenced in the third-order quantum polarization behavior shown in Fig.~\ref{fig:fig4}(d). Note also from Fig.~\ref{fig:fig4}(d) that the N00N state offers three-fold improvement of phase sensitivity over the classical behavior as well as the possibility to get unity interference visibility. Such information is not at all evident from the density matrix description of the quantum states shown in Fig.~\ref{fig:fig3}, although, for a two-mode state with $N$-photons, or equivalent, a state of composite system for $N$ spin-$1/2$ particles, the two figures contain mathematically equivalent and interconvertible information. Additionally, deviations of the experimental quantum states from their ideal target states are more easily identified in Fig.~\ref{fig:fig4} than in Fig.~\ref{fig:fig3} as it is difficult to deduce such information from a visual inspection of the density matrices in Fig.~\ref{fig:fig3}.
\section{\label{sec:5}Conclusion}
We have experimentally studied diverse quantum polarization features of different three-photon states, selected to represent the six possible three-photon polarization classes. The states have interesting characteristics such as perfect polarization, absence of polarization, hidden polarization and maximum sum-uncertainty. Our classification and experimental results for three-photon states can be applied as well to describe the spin features of composite systems consisting of three spin-1/2 particles with the bosonic characteristic. In addition, we have shown that subtle quantum polarization features are more sensitive to state imperfections than those of state density matrices. Our results hint that the central moment description can be used to describe the quality of a multi-photon polarization state with better sensitivity, in particular in the cases where higher order polarization features are important.
\
\section*{Acknowledgments}
This work was supported by the National Research Foundation of Korea (Grant Nos. 2016R1A2A1A05005202, 2016R1A4A1008978, and NRF-2012K1A2B4A01033433). Y.K. acknowledges support from the Global Ph.D. Fellowship by the National Research Foundation of Korea (Grant No. 2015H1A2A1033028). G.B. acknowledges financial support by Swedish Foundation for International Cooperation in Research and Higher Education (STINT), the Swedish Research Council (VR) through its Linnaeus Center of Excellence ADOPT and Contract No. 621-2014-5410.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,428 |
{"url":"https:\/\/electronics.stackexchange.com\/questions\/210064\/calculation-of-v-o-v-i\/413625","text":"# Calculation of $V_o\/V_i$\n\nI know this might be a bit too specific of a question, but I have trouble calculating $V_o\/V_i$ in the given scheme.\n\nI was given a BJT-transistor and had to find an equivalent AC-scheme (i.e. eliminating conductors and DC sources).\n\nNow, I am asked to calculate $V_o\/V_i$ as seen in the image.\n\nCan anyone give some tricks where to start etc. and\/or perhaps give a solution? There are no numerical values, so the solution should be in 'symbolic' form. The part which confuses me is the 'current controlled current source' and the resistor $R_B$ (more precise, how to bring it in to account).\n\n\u2022 Where's i_B in the circuit? \u2013\u00a0Andy aka Jan 7 '16 at 21:14\n\u2022 Between the top two left points, left to right (is that clear, else I'll make a new picture) \u2013\u00a0Robin Haveneers Jan 7 '16 at 21:28\n\u2022 The equivalent diagram represents one of the classical BJT amplifier schemes: Common emitter with voltage controlled current feedback (Resistor RB). The calculation would not too complicated in case RB>> Rc||Rl. However, without this simplification, it would be rather involved. \u2013\u00a0LvW Jan 7 '16 at 21:31\n\u2022 The exercise states \"Let's assume the transistor is correctly set up with $V_{cc}, R_b, R_c$ . Not really sure what that's supposed to mean.. ( $V_{cc}$ obviously left out because this is the AC-equivalent.) \u2013\u00a0Robin Haveneers Jan 7 '16 at 21:35\n\u2022 R_B is a feedback resistor. I remember generally how to solve these problems, but haven't done it in ages. You start by opening the feedback loop. Determine the open-loop gain (gain with R_B open). Then you need to determine the feedback type and feedback factor. Then use feedback factor to calculate the closed loop gain. Wikipedia might help. en.wikipedia.org\/wiki\/Negative_feedback_amplifier \u2013\u00a0mkeith Jan 7 '16 at 23:10\n\n\"Can anyone give some tricks where to start etc\"\n\nOK - here is a rough description of my approach.\n\n\u2022 At first calculate the gain Ao from base to collector using the known formula transconductance x resistance Rtot (total at the collector): Ao=-gm*Rtot. Transconductance gm=beta\/Rpi.\n\n\u2022 Rtot=Rc||RL||Rbc with Miller-resistor at the collector node Rbc=Rb\/(1-1\/Ao), Ao negative!\n\n\u2022 Hence: Ao=-gm\/(1\/Yc)+gm\/(1\/YL)+gm*(1-1\/Ao)\/(1\/Yb) . This equation must be solved for Ao (Ao appears twice!).\n\n\u2022 As the last step, you calculate the input resistance Rin=Rpi||Rbb at the base node using the Miller-Theorem (at the base) with Rbb=Rb\/(1-Ao).\n\n\u2022 Finally, you can compute the resulting gain A=Ao*Rin\/(Rin+Rs)\n\nEDIT_1 (comment): Vo\/Vi=Ao because Vi=V at the base node.\n\nEDIT_2: Because you specifically ask for the exact role of Rb:\n\nThe classical procedure to calculate the output voltage Vo leads to a negative value (with respect to Vi) because the gain Ao is inverting. However, there is another unwanted part of Vo which is NOT inverted. This part is not yet included in my above calculation. This part of the output voltage arrives DIRECTLY at the collector node (not via the transistors collector current) and is found using the simple voltaged divider rule Vi*Rtot\/(Rtot+Rb). As you can see, this (positive) part disappears for Rb>>Rtot.\n\nAlthough conventions for currents and voltages are arbitrary indeed, I prefer looking at a circuit of which I 'understand' and feel familiar with the notations and polarities. Therefor I'd like to redraw the circuit, defining the BJT's base current as $$\\ i_b \\$$, replacing $$\\ R_B \\$$ and $$\\ i_B \\$$ with $$\\ R_{fb} \\$$, and $$\\ i_{fb} \\$$ respectively, as they pertain to feedback. In the end these choices will not change the outcome $$\\ \\displaystyle \\frac{V_o}{V_i} \\$$.\n\nsimulate this circuit \u2013 Schematic created using CircuitLab\n\nWith Eric Best's approach $$i_{fb} + i_{R_o} + \\beta i_b = 0$$ hence $$V_o \/ R_o + (V_o-V_i)\/R_{fb} + \\beta V_i\/R_\\pi = 0$$\n\nor\n\n$$V_o \\left( \\frac{1}{R_o} + \\frac{1}{R_{fb}} \\right) = V_i \\left( \\frac{1}{R_{fb}} - \\frac{\\beta}{R_\\pi} \\right)$$\n\nand indeed\n\n$$\\frac{V_o}{V_i} = \\displaystyle \\frac{\\frac{1}{R_{fb}} - \\frac{\\beta}{R_\\pi}}{\\frac{1}{R_c} + \\frac{1}{R_l} + \\frac{1}{R_{fb}}}$$ which after replacing $$\\ R_{fb} \\$$ back with $$\\ R_B \\$$ becomes $$\\frac{V_o}{V_i} = \\displaystyle \\frac{\\frac{1}{R_B} - \\frac{\\beta}{R_\\pi}}{\\frac{1}{R_c} + \\frac{1}{R_l} + \\frac{1}{R_B}}$$ which is exact confirmation of Eric Best's result.\n\n\u2022 But what if you want to find Vo\/Vs instead? \u2013\u00a0G36 Dec 25 '18 at 7:31\n\u2022 @joe electro Yes, very good. In my solution I was just keeping the attitude\/conventions of the original question. In one of my comments I wrote \"it is more usual, in my opinion, to choose the opposite polarity than that chosen in the diagram.\". Your solution fully corresponds with common conventions. \u2013\u00a0Eric Best Dec 25 '18 at 13:21\n\u2022 @G36 See my EDIT (continuation of my original answer), where you can find my solution concerning Vo\/Vs. \u2013\u00a0Eric Best Dec 25 '18 at 21:26\n\nThe current controlled current source (CCCS) represents the amplification property of the BJT and in this case, it means that the current through this source is given as $$\\ \\beta \\cdot I_b \\$$ (as it is stated in the diagram, $$\\ I_b \\$$ being the BJT's base current). However, realize that $$\\ I_b \\$$ is the current flowing only through $$\\ R_\\pi \\$$ (downwards - this is determined by given relation to the direction of the current through CCCS drawn in the diagram) because $$\\ R_B \\$$ is an external element (not part of the BJT)! See the Andy aka's question \"Where's i _ B in the circuit?\" and the Robin Haveneers's answer \"Between the top two left points, left to right\" - which must be wrong from the reason mentioned above.\n\nSo, this base current $$\\ \\displaystyle I_b \\$$ is given as $$\\ \\displaystyle -V_i \/ R_\\pi \\$$ (I have applied the $$\\ \\displaystyle V_i \\$$ polarity used in the diagram, the same later below with the $$\\ V_o \\$$ polarity).\n\nLet's define $$\\ R_o = R_c || R_l \\$$ (just to simplify it a bit)\n\nThen we can write:\n\n1. For the current flowing through $$\\ R_B \\$$ (to the left; its direction is arbitrarily chosen by me):\n\n\u2022 $$\\ \\displaystyle I_{R_B}=\\frac{V_i-V_o}{R_B} \\$$.\n2. For the current flowing through $$\\ R_o = R_c||R_l \\$$ (upwards and to the left, its direction is arbitrarily chosen by me again):\n\n\u2022 $$\\ \\displaystyle I_{R_o} = \\frac{V_o}{R_o} \\$$.\n3. For the current flowing through the CCCS (downwards, already stated above, its direction is determined in the diagram):\n\n\u2022 $$\\ \\displaystyle I_{CCCS}=\\beta \\cdot I_b = -\\beta \\cdot \\frac{V_i}{R_{\\pi}} = -g_m \\cdot V_i \\$$,\n\nthe transconductance $$\\ g_m \\$$ being defined as $$\\ \\displaystyle \\frac{\\beta}{R_\\pi} \\$$.\n\nSince according to KCL: $$\\ \\displaystyle I_{R_o}=I_{R_B}+I_{CCCS} \\$$, we obtain:\n\n$$\\ \\displaystyle \\frac{V_o}{R_o} = \\frac{V_i-V_o}{R_B} - \\beta \\cdot \\frac{V_i}{R_{\\pi}} \\$$,\n\nand, after some rearrangement:\n\n$$\\ \\displaystyle \\frac{V_o}{R_o} + \\frac{V_o}{R_B} = \\frac{V_i}{R_B} - \\beta \\cdot \\frac{V_i}{R_{\\pi}} \\$$\n\nFrom here it is already very easy to express the required gain:\n\n$$\\ \\displaystyle A_V = \\frac{V_o}{V_i} = \\frac{\\frac{1}{R_B} - \\frac{\\beta}{R_{\\pi}}} {\\frac{1}{R_B} + \\frac{1}{R_o}} = \\frac{\\frac{1}{R_B} - \\frac{\\beta}{R_{\\pi}}} {\\frac{1}{R_B} + \\frac{1}{R_c} + \\frac{1}{R_l}} = \\frac{\\frac{1}{R_B} - g_m} {\\frac{1}{R_B} + \\frac{1}{R_c} + \\frac{1}{R_l}} \\$$\n\nEDIT (Dec 25 2018):\n\nIf we want to know the relation between $$\\ V_o \\$$ and $$\\ V_s \\$$ (as G36 mentioned in his comment), we can continue the following way:\n\nUsing KCL, we can write: $$\\ \\displaystyle I_{R_B} = I_b + I_{R_S} \\$$ (with $$\\ I_{R_S} \\$$ flowing downwards; chosen by me)\n\nFurther, using the voltage polarities from the original diagram (notice the $$\\ V_s \\$$ polarity in relation to $$\\ V_i \\$$), we obtain:\n\n$$\\ \\displaystyle \\frac{V_i - V_o}{R_B} + \\frac{V_i}{R_\\pi} + \\frac{V_i + V_s}{R_S} = 0 \\$$\n\nWe already know the relation between $$\\ V_o \\$$ and $$\\ V_i \\$$ ($$\\ A_V \\$$), so it's easy to express either the \"overall\" gain $$\\\\displaystyle \\frac{V_o}{V_s} \\$$ or just the partial one $$\\\\displaystyle \\frac{V_i}{V_s} \\$$, substituting for either $$\\ V_i \\$$ or for $$\\ V_o \\$$ in the above equation and rearranging it to get the required voltage ratio:\n\n1. $$\\ \\displaystyle V_i = \\frac{V_o}{A_V} \\$$\n\n$$\\ \\displaystyle \\frac{V_o}{V_s} = \\frac{1}{\\frac{R_S}{R_B} - \\frac{1}{A_V} \\cdot [1 + R_S (\\frac{1}{R_B} + \\frac{1}{R_\\pi})]} \\$$\n\n2. $$\\ \\displaystyle V_o = {A_V}{V_i} \\$$\n\n$$\\ \\displaystyle \\frac{V_i}{V_s} = -\\frac{1}{1 + R_S (\\frac{1}{R_\\pi} + \\frac{1 - A_V}{R_B})} \\$$\n\n(Notice: $$\\ V_s \\$$ is a voltage of an ideal voltage source (thus it's the internal voltage of it) and $$\\ R_S \\$$ evidently represents here the internal resistance of that source. It determines an attitude to the above derived formulas - I understand the only reachable\/accessible voltage is $$\\ V_i \\$$ in fact.)\n\n\u2022 I suggest to write $R_l$ instead of $R_I$, as my guess is that $R_l$ is the load's resistance. In the same line I'd suggest to use $R_o$ ('o' for 'out') instead of $R_{tot}$; it's shorter and me seems more to the point. Just some minor suggestions, still going through your calculations. I also assume the current $I_{Rb}$ is flowing to the right, instead of to the left. \u2013\u00a0joe electro Dec 24 '18 at 5:17\n\u2022 If you define $I_{R_{tot}}$ upward and to the left, it should read $-\\frac{V_o}{R_{tot}}$. Regarding point 3.: $I{CCCS}$ I'm getting confused now. Please illustrate your answer with a drawing indicating the polarities and current directions. \u2013\u00a0joe electro Dec 24 '18 at 5:21\n\u2022 May I inquire your voltage polarity convention? Is the positive polarity located at the tail of the arrow? Because that would lead to $I_b = \\frac{V_i}{R_b}$ if $I_b$ is defined downward through $R_\\pi$. Otherwise I think it should be that $I_b = - \\frac{V_i}{R_b}$. Also assuming that with 'b' (small capital) you mean the BJT's base, and with 'B' you mean the feedback resistor. \u2013\u00a0joe electro Dec 24 '18 at 5:29\n\u2022 @joe electro I have made some changes according to your advice (I -> l, tot -> o), but: 1. You wrote: \"I also assume the current $I_{R_b}$ is flowing to the right, instead of to the left.\" - No, the current direction can be chosen arbitrarily, what I did. The result direction is given by its sign (if positive, then it is identical with the chosen one, if negative, then it is opposite). (to be continued) \u2013\u00a0Eric Best Dec 24 '18 at 21:44\n\u2022 @joe electro (cont.) 2. You wrote: \"If you define $I_{R_{tot}}$ upward and to the left, it should read $\u2212\\frac{V_o}{R_{tot}}$.\u201c \u2013 No again, have a look at the polarity of $V_o$ in the diagram (it means $V_o$ is located at the lower connection of the resistors in fact, being referenced to the higher connection of them). It is a convention that the arrow points from a point with a \u201ehigher voltage\u201c (or potential, if you like) to a point with a lower one. (to be continued) \u2013\u00a0Eric Best Dec 24 '18 at 21:45\n\nUse KCL\/KVL. There are only two unknown currents, so just need two meshes for KVL. The current source should not be in either of those two meshes. BTW, are you sure that $I_B$ does not flow through $R_\\pi$?\n\n\u2022 I agree with Chu - the current $I_B$ flows for sure only through $R_\\pi$ - this is the actual base, $R_b$ is an external element! \u2013\u00a0Eric Best Dec 24 '18 at 0:31","date":"2019-05-25 23:08:46","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\": 55, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8299561738967896, \"perplexity\": 948.2931188504639}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232258453.85\/warc\/CC-MAIN-20190525224929-20190526010929-00415.warc.gz\"}"} | null | null |
\section{Introduction}
Reaction-diffusion processes in the presence of quenched disorder (i.e. traps) has been studied extensively for almost a hundred years. Smulochowsky introduced a reaction diffusion system with traps as a model to study the process of coagulation in colloids\cite{smolu1916}. Reaction diffusion process with traps has been used to study a number of phenomena such as exciton trapping in crystals, recombination of electro-hole, soliton-antisoliton pair, reaction activated by catalysts etc\cite{hav}. Quenched disorder can introduce self-segregation in the reaction process which affects the reaction kinetics\cite{jayan,htait}. It can also induce self-organization of reactant species around the traps\cite{wei89,tait,jayan1}. It has been shown by Balagurov and Vaks \cite{balagurov}, Donsker and Vardhan\cite{donsker} and Grasberger and Procaccia\cite{gras} that the mean survival probability in the asymptotic time limit has a stretched exponential behavior $P(t) \sim \exp(-\alpha_d \rho^{2/(d+2)}t^{d/(d+2)})$, where $\alpha_d$ is a constant that depends on the dimensionality of the space $d$. This slowing down behavior arises due to the occurrence of large trap free regions in space. If traps are located at regular interval the survival probability shows an exponentially decreasing behavior\cite{wi}. It has also been shown that diffusion in the presence of random traps cannot be described by effective diffusion equation\cite{bixon}.
A reaction diffusion process with traps can be described as $A+T \rightarrow T$ with a rate $\kappa$ where $T$ and $A$ denotes trap and the diffusion particle respectively. For the case of perfect traps we shall consider the limit $\kappa \rightarrow \infty$ for which every particle arriving at the trap vanishes with probability $1$. The case where $\kappa$ is finite is called trapping reaction with partial traps\cite{thm,wi}. In one dimension the mean survival probability in the asymptotic time limit is given by
\begin{equation}
\begin{aligned}
P(t)= 16\pi^{-3/2}\exp \left(\alpha (Dt)^{1/3}\right)\left[ b \sqrt{Dt} + c(Dt)^{1/6}+O((Dt)^{-1/6})\right]
\label{surv_prob}
\end{aligned}
\end{equation}
where $\alpha= -3\rho^{2/3} \pi^{2/3}2^{-2/3}$, $b=2 \pi \rho 3^{-1/2}$ and $c=17 \pi^{1/3}\rho^{1/3}2^{-1/3}3^{-1/2}9^{-1}$. Here $D$ is the diffusion constant of the reacting particles and $\rho$ is the average number of traps per unit length. Exact expression for $P(t)$ can be obtained in terms of Meijer G functions. This slowing down behavior is observed due to the existence of large trap free regions with small but nonzero probabilities.
In previous studies mutual interaction such as exclusion between particles has not been considered. In this article, we shall investigate the effect of exclusion on the asymptotic survival probability. In a reaction diffusion system the reactants are usually assumed to be point particles. This assumption is valid when volume exclusion it not important i.e. in diffusion processes where the concentration low. However if we consider volume exclusion in the reaction diffusion process with traps, the stretched exponential law gets modified. In our investigation we have found that an additional correction term $\sim t^{1/6}$ appear in the exponent. This is also verified by stochastic simulations.
Let us denote by $u(x,t)$ the density of the reactant at position $x$ at time $t$. Due to the repulsive nature of exclusion smoothing of density is favored. Reactants particle moving towards a local maxima in the density will experience a local drift away from the local maxima. This repulsive force which arises due to exclusion can be assumed to be proportional to the gradient $\partial_{x} u(x,t)$. We introduce this as a small perturbation in the reaction diffusion equation. The reaction diffusion equation can be written as
\begin{equation}
\partial_t u = \partial_{x} (D \partial_{x} u + \epsilon(\partial_{x} u) u))+\kappa \sum_{i}\delta(x-x_i)u,
\label{diffeqn2}
\end{equation}
where $x_1,x_2\ldots$ denotes the positions of the traps which are uniformly distributed with mean number of traps per unit length $\rho$. The boundary condition is $\partial_x u =0$ at infinity. The parameter $\epsilon$ is a positive constant that determines the strength of exclusion. We note here that exclusion in \eq{diffeqn2} has been introduced in a meanfield way.
\section{Survival probability}
For the case of perfect traps i.e. limit $\kappa \rightarrow \infty$ the problem reduces to solving the diffusion equation \eq{diffeqn2} in a finite domain $[-a, a]$ with absorbing boundary conditions. We have
\begin{equation}
\partial_t u = \partial_{x}((1+\epsilon u) \partial_{x} u),~~x \in [-a, a], t>0,
\label{diff_eqn}
\end{equation}
where diffusion constant $D$ is assumed unity. The initial concentration $u(x,0)=1$ and the absorbing boundary condition is $u(\pm a, t)=0 \mbox{~for all~} t>0$. The exponent $~t^{1/3}$ in \eq{surv_prob} has been calculated by taking into account only the lowest mode of the eigenvalue equation $D \partial_{x}^2 \psi + \lambda \psi =0$. In the presence of exclusion we have the following nonlinear eigenvalue problem
\begin{equation}
\partial_{x}((1+\epsilon \psi) \partial_{x} \psi)+\lambda \psi =0,~~x \in [-a, a].
\label{egv_eqn}
\end{equation}
To understand the behavior of the survival probability for we need to know how the eigenvalue $\lambda$ get modified for nonzero $\epsilon$. For small $\epsilon$ we can write
\begin{eqnarray}
\psi = \psi_0 + \epsilon \psi_1 + \epsilon^2 \psi_2 + \ldots, \nonumber \\
\lambda = \lambda_0 + \epsilon \lambda_1 + \epsilon^2 \lambda_2 + \ldots.
\label{pert_ser}
\end{eqnarray}
Substituting \eq{pert_ser} in \eq{egv_eqn} we obtain
\begin{subequations}
\begin{align}
\label{eq1}
O(1)&:~~\partial_{x}^2\psi_0+\lambda_0 \psi_0 =0,\\
\label{eq2}
O(\epsilon)&:~~\partial_{x}^2\psi_1+\lambda_0 \psi_1 =-\lambda_1 \psi_0-\partial_{x}(\psi_0 \partial_{x} \psi_0).
\end{align}
\end{subequations}
The above set of linear equation can be solved with the absorbing boundary conditions $\psi_m(\pm a)=0,~~m=0,1,2,\ldots$. The eigenvalues and the eigenfunctions of \eq{eq1}are given by
\begin{eqnarray}
\label{sol_pert0}
\psi_0^{(n)}(x) &=& \frac{1}{\sqrt{a}} \left \{ \begin{array}{ll}
\cos(n\pi x/2a), & \mbox{if $n$ is odd}, \\
\sin(n \pi x/2a), & \mbox{if $n$ is even},
\end{array} \right. \\
\label{sol_pert1}
\lambda_0^{(n)} &=& \frac{n^2 \pi^2}{4a^2}~~\mbox{for}~ n=1,2,\ldots
\end{eqnarray}
Similarly, from \eq{eq2} after setting $\lambda_0 = \lambda_0^{(n)}$ and $\psi_0 = \psi_0^{(n)}$ we have
\begin{equation}
\partial_{x}^2 \psi_1 + \lambda_0^{(n)} \psi_1 = -\lambda_1 \psi_0^{(n)} - \partial_{x}(\psi_0^{(n)} \partial_{x} \psi_0^{(n)}).
\label{eq_ord1}
\end{equation}
Let $\psi_1$ be written as the sum $\psi_1 = \sum_m A_m \psi_0^{(m)}$ which satisfies the boundary condition. From \eq{eq_ord1} we obtain
\begin{equation}
A_m(\lambda_0^{(n)}-\lambda_0^{(m)}) = -\lambda_1 \delta_{m,n}-\int_{-a}^{+a}\partial_{x}(\psi_0^{(n)} \partial_{x} \psi_0^{(n)})\psi_0^{(m)} dx
\label{pertb_A}
\end{equation}
Using \eq{sol_pert0}, \eq{sol_pert1} and \eq{pertb_A} we can write for odd $n$
\begin{equation}
\int_{-a}^{+a}\partial_{x}(\psi_0^{(n)} \partial_{x} \psi_0^{(n)})\psi_0^{(m)} dx = -\frac{\lambda_0^{(n)}}{\sqrt{a}}\mu_{n,m}
\label{expr_mu}
\end{equation}
where $\mu_{n,m} = [4m \cos(n \pi) \sin (m \pi/2)-8n \cos(m \pi/2) \sin (n\pi)]/\pi (m^2-4n^2)$ for $m$ odd otherwise $\mu_{n,m} = 0$. Similarly for even $n$ we can calculate $\mu_{n,m}$. Here we will not require to calculate beyond $\mu_{1,1}$ since we only need $\lambda^{(1)} = \lambda_0^{(1)}+ \epsilon \lambda_1^{(1)}$ to determine the behavior of the asymptotic survival probability. The eigenfunctions and the eigenvalues for $n=1$ can be written as
\begin{eqnarray}
\psi^{(1)} &=& \psi_0^{(1)} +\frac{\epsilon}{\sqrt{a}} \sum_{m \neq 1}^{\infty} \mu_{1,m} \frac{\lambda_0^{(1)}}{\lambda_0^{(1)}-\lambda_0^{(m)}} \psi_0^{(m)}, \nonumber \\
\lambda^{(1)} &=& \lambda_0^{(1)} \left(1+\frac{\epsilon \mu_{1,1}}{\sqrt{a}} \right).
\end{eqnarray}
The greatest contribution to the stretched exponential behavior comes from $n=1$ case. The survival probability can therefore be written as
\begin{equation}
P_{\epsilon}(t) \sim \int_{0}^{\infty} \exp(-\lambda^{(1)}(x)t) \rho^2 4x \mbox{e}^{-2 \rho x}dx
\label{prob_surv_eps}
\end{equation}
We will use the Laplace method to evaluate \eq{prob_surv_eps} as done by Balagurov and Vaks \cite{balagurov}. The exponent in \eq{prob_surv_eps} can be explicitly written as
\begin{equation}
f(x) = -\frac{\pi^2}{x^2}\left(1+ \frac{\sqrt{2}\epsilon \mu_{1,1}}{\sqrt{x}} \right)t-\rho x.
\end{equation}
The function $f(x)$ attains maximum at
\begin{equation}
\tilde{x} \simeq x^{*}+\epsilon \frac{5 \mu_{1,1}}{6 \sqrt{2}} \sqrt{x^{*}}
\end{equation}
where $x^{*} = (2 \pi^2 t/\rho)^{1/3}$. The integral \eq{prob_surv_eps} becomes
\begin{equation}
\begin{aligned}
P_{\epsilon}(t) &\sim \alpha \exp(f(\tilde{x})),\\
&= \alpha \exp \left(\frac{-3(\rho^2 \pi^2 t)^{1/3}}{2^{2/3}} - \epsilon \mu_{1,1} \left( \frac{\pi}{2} \right)^{1/3} \rho^{5/6} t^{1/6} \right),
\end{aligned}
\label{prob_excl}
\end{equation}
where $\alpha=\left(\frac{\pi}{2 f''(\tilde{x})}\right)^{1/2}\tilde{x},~\mu_{1,1} = 4/3\pi$.
\section{Numerical simulation and results}
Numerical solution of the diffusion equation \eq{diff_eqn} is obtained by the Langevin simulation of interacting particle system. In our simulation we cannot introduce exclusion between pair of particles by a repulsive interaction. As this is not a lattice simulation where exclusion is incorporated by fixing the upper bound of the site occupancy. The difficult arises here since we have a mean-field description of repulsive interaction in \eq{diff_eqn}. Therefore we need to solve the problem in a self-consistent manner. We consider $N$ particles in $[-a,a]$ with a uniform density $u(x,t)=1$ at time $t=0$. A corresponding self-consistent Langevin equation can be written as
\begin{equation}
X^{(i)}_{t+dt} = X^{(i)}_t + F(X^{(i)}_t)dt + \sqrt{2Ddt} \xi^{(i)}_t, i=1,2,\ldots N
\label{langv}
\end{equation}
with the force term $F(X^{(i)}_t) = -\epsilon \partial_{x} u(x,t)|_{x=X^{(i)}_t}$ and initial conditions $X^{(i)}_0=\zeta_i$ is a uniform random variable in $[-a,a]$ such that the initial density $u(x,0)$ is constant. The diffusion constant $D=1$ and $\xi^{(i)}_t \sim N(0,1)$ is a Gaussian random variable.
Evolution of the $N$ particle system is performed self consistently in the following way. In the interval $[-a,a]$ $N$ particles are uniformly distributed initially. Let $x_1,\ldots, x_M$ denote a partition of the interval $[-a,a]$. The density $u(x,0)$ is calculated at $x_i$ for all $i=0,1,\ldots,M$. The system is evolved using \eq{langv} for time step $dt$ and the density is updated simultaneously. The force term $F(X^{(i)}_t)$ is approximated by using central difference for the gradient$\partial_{x} u(x,t)|_{x=X^{(i)}_t} \simeq (u(x_{n+1},t)-u(x_{n-1},t))/2dx$ where $X^{(i)}_t \in [x_n-dx/2,x_n+dx/2]$. The absorbing boundary conditions $u(\pm a,t)=0$ is incorporated by eliminating each particle that exits the interval $[-a,a]$. The system is evolved for a number of times from the same initial condition. The mean density obtained should satisfies the diffusion equation \eq{diff_eqn} and is the required solution. In \fig{fig:sol} we compare the solution obtained from the numerical simulation for $\epsilon=0$ with the exact solution and find that they are in excellent agreement.
From the analytical expression in \eq{prob_excl} we have
\begin{equation}
\log \left| \log \frac{P_\epsilon(t)}{P_0(t)} \right| = \frac{1}{6} \log t + C,
\label{slope_eq}
\end{equation}
where $C$ is constant. Note that this is true only for small values of $\epsilon$ i.e. $0<\epsilon \ll 1$. The results from the simulation is shown in \fig{fig:slope}. Here we have used the numerical solution to calculate the survival probability $P_\epsilon(t)$. We have taken values $\epsilon = 0.2$ and $0.4$, the diffusion coefficient $D=1$ and the mean number of traps per unit length $\rho$ is unity. Result from our simulation were fitted with straight lines by least square fitting for $\log t \geq 1$. In \fig{fig:slope} the slopes of the straight lines are $0.1784\pm0.0070$ and $0.1831\pm0.0047$ for $\epsilon = 0.2$ and $0.4$ respectively. Although values obtained are slightly higher than the predicted value $0.166\ldots$, the results are satisfactory. We note that the slope $1/6$ is obtained by first order perturbation in $\epsilon$ and it is difficult to obtain an analytical expression with higher order perturbation. For higher values of $\epsilon$ higher order terms will become important in the stretched exponent. However we note that the term $\epsilon \partial_x u$ in \eq{diffeqn2} arising due to exclusion effect is valid for small $\epsilon$.
\section{Conclusion}
We investigated the trapping reaction problem in the presence of exclusion where we found that the survival probability gets modified. We obtained analytically, additional correction term in the stretched exponential law for the survival probability. Our analytical prediction is verified by numerical simulations. Self-consistent Langevin simulation is used to investigate nonlinear reaction diffusion equations. The algorithm used can be further explored to investigate trapping problems in higher dimensions and other (in)homogeneous reaction-diffusion systems with multiple species of reactants.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,482 |
Native American Salary Loan "Wwwwirepayday Com Login". After you have spoken with family members and friends potentially taking out a short-term loan, and they do not have the money to lend you, you might want to consider other options, one of which is a payday loan company, a business that is designed to help people that are in these situations. You could go to a credit union or a bank in an attempt to get a similar unsecured loan, but unless you have an account with them, such as with the mortgage, it is unlikely that they will grant your request. If you do not have a credit card where you can take money out as in advance, you will probably want to work with a payday loan company. Wire Payday bad credit payday loans is a company that is specifically therefore people that have low credit scores. If this is reflective of your situation, the following information will help you understand why this might be the exact company that you need to work with trade. You can get cash loans for fair credit by using Wwwwirepayday Com Login, and read reviews.
Looking for Wwwwirepayday Com Login. Online pay day loans $100 to be able to $1000. Easy Credit assessment Fax less. 99% Acceptance Secure Software. Understand it Today.
There are some people that will show you that the payday advance company is not the perfect option to create if you are searching for a quick-term loan because of the interest rates they charge. Their business not understand is the fact that by missing certain payments you could find yourself paying certain fees and penalties which could add up to far more than you would ever pay with one of these short-term lenders. In reality, it's a much better idea to take into consideration employing a business similar to this because you should have a high probability of getting the money that you require very quickly. Some of them actually boast about providing money within 24 hours that you requested, and this will help to you receive your money back in line.
You will pay back these loans quickly, usually within a couple of weeks, and it will surely be paid back using the paycheck which you have to arrive at that time. This money is what you will use to pay for the loan off, and thus you must never go over that amount while you are getting a loan from one of these simple businesses. You should also consider just how much interest will accrue during this period, which details are typically presented of the paperwork that you just will sign whenever you finally have the money. Once you have calculated how much you require, and the way much your following paycheck will probably be, you can develop the right amount.
It is advisable to submit this application online since this is the easiest way of getting it towards the firms that can supply you with the cash. Many of them do have physical locations that you can stop in, and do the exact same thing, however, many times they are certainly not as quickly. It's also much easier to simply get on your laptop computer, or desktop PC, and submit the info over the web. When you receive your approval, these funds will probably be deposited rapidly into the banking account, enabling you to obtain the money that you need to pay your debts. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,226 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.